- Stationary waves are formed by the superposition of two identical waves travelling in opposite directions.
- In a stationary wave, some points remain at rest (nodes), and some vibrate with maximum amplitude (antinodes).
- The distance between two consecutive nodes or two consecutive antinodes is \[\frac {λ}{2}\], and between a node and adjacent antinode is \[\frac {λ}{4}\].
- All particles between two adjacent nodes vibrate in the same phase, but adjacent loops vibrate in opposite phase.
- In stationary waves, energy is not transferred; the wave is localised, and the wave velocity is zero.
Definitions [34]
Definition: Superposition of Waves
The phenomenon in which, when two or more waves arrive at a point simultaneously, each wave produces its own displacement independent of the other, and the resultant displacement at a point is the vector sum of the displacements due to the individual waves, is called superposition of waves.
Definition: Progressive Wave
A wave in which the disturbance produced in the medium travels in a given direction continuously, without any damping and obstruction, from one particle to another, is called a progressive wave.
Definition: Transverse Wave
A wave in which vibrations of particles are perpendicular to the direction of propagation of the wave and produce crests and troughs in their medium of travel is called a transverse wave.
Definition: Longitudinal Wave
A wave in which vibrations of particles produce compressions and rarefactions along the direction of propagation of the wave is called a longitudinal wave.
Definition: Reflection of Waves
The phenomenon that occurs when a progressive wave travelling through a medium reaches a rigid boundary and gets reflected is called reflection of waves.
Definition: Stationary Wave
When two identical waves travelling along the same path in opposite directions interfere with each other, the resultant wave is called a stationary wave.
Definition: Forced Vibration
The vibration that occurs due to a continuous external force acting on the system, where the body vibrates at the frequency of the external periodic force, is called forced vibration.
Definition: Free Vibration
The vibration that occurs without any external periodic force after an initial disturbance, where the body vibrates at its own natural frequency, is called free vibration.
Definition: Harmonics
All the frequencies that are integral multiples of the fundamental frequency are called harmonics.
Definition: Overtones
Only those multiples of fundamental frequency which are actually present in a given sound are called overtones.
Definition: Organ Pipe
The musical instruments which are used for producing musical sounds by blowing air into them are called organ pipes.
Definition: Waxing
The maximum intensity point produced during the formation of beats is called waxing.
Definition: Waning
The minimum intensity point produced during the formation of beats is called waning.
Definition: Beats
The variation in intensity of sound with time at a particular position, due to the principle of superposition of two sound waves of slightly different frequencies, is called beats.
Definition: Phenomenon of Beats
The periodic variation of intensity of sound between maximum and minimum due to superimposition of two sound waves of same amplitude and slightly different frequencies is called the phenomenon of beats.
Define the following term:
Frequency
The frequency of a particle executing S.H.M. is equal to the number of oscillations completed in one second.
Definition: Wind Instruments
The musical instruments that consist of an air column in which sound is produced by setting vibrations of the air column are called wind instruments. (Examples: Mouth organ, flute)
Definition: Stringed Instruments
The musical instruments that consist of a stretched string in which sound is produced by plucking of strings are called stringed instruments. (Examples: Tanpura, sitar, guitar)
Definition: Percussion Instruments
The musical instruments in which sound is produced by setting vibration in a stretched membrane are called percussion instruments. (Examples: Tabla, drum)
Definition: Principle of Superposition
When two or more waves, travelling through medium, pass through common point, each wave produces its own displacement at that point, independent of the presence of the other wave. The resultant displacement at that point is equal to the vector sum of the displacements due to the individual wove at that point.
Definition: Resonance
The phenomenon in which the amplitude of forced vibrations becomes maximum when the driving frequency equals the natural frequency.
Definition: Reflection of Waves
When a progressive wave, travelling through a medium, reaches an interface separating two media, a certain part of the wave energy comes back in the same medium. The wave changes its direction of travel. This is called reflection of a wave from the interface.
Definition: Forced Vibrations
Vibrations in which a body vibrates under the influence of an external periodic force, with the frequency of the external force.
Definition: Fundamental Frequency
The lowest natural frequency of a vibrating system is called the fundamental frequency or first harmonic.
Definition: Fundamental Mode
The mode of vibration corresponding to the fundamental frequency is called the fundamental mode or fundamental tone.
Definition: Sound Box
The hollow wooden box used to amplify the sound produced by the vibrating string is called the sound box.
Definition: Mechanical Wave
A mechanical wave is a disturbance produced in an elastic medium due to periodic vibrations of particles of the medium about their respective mean positions.
Definition: Natural Frequency
The frequency at which a body vibrates when disturbed and left free to vibrate on its own.
Definition: Free Vibrations
Vibrations in which a body vibrates at its natural frequency without any external periodic force.
Definition: Sonometer
A sonometer is an apparatus used to study the vibrations of a stretched string and to verify the laws of a vibrating string.
Definition: Progressive Wave
A wave, in which the disturbance is produced in the medium travels in a given direction continuously, without any damping and obstruction, from one·particle to another, is a progressive wave or a travelling wave.
Definition: Beats
Beats are the periodic variation in intensity of sound produced due to the superposition of two sound waves having nearly equal (slightly different) frequencies.
Definition: Time Period of Beats
The interval between two maximum sound intensities is the time period of beats.
Definition: Stationary Wave
When two identical waves travelling along the same path in opposite directions interfere with each other, resultant wave is called stationary wave.
Formulae [10]
Formula: Equation of Stationary Wave
y = 2A cos\[\left(\frac{2\pi x}{\lambda}\right)\sin(2\pi nt)\]
In terms of resultant amplitude R: y = R sin(2πnt)
where: R = 2A cos\[\left(\frac{2\pi x}{\lambda}\right)\]
Formula: Displacement Relation of a Progressive Wave
y(x,t) = a sin(kx − ωt + ϕ0)
where:
- a = Amplitude
- k = Angular wave number
- ω = Angular frequency
- ϕ0 = Initial phase
If displacement is a linear combination of sine and cosine:
y(x,t) = A sin(kx − ωt) + B cos(kx − ωt)
Formula: Closed Organ Pipe
| Mode | Formula | Also Known As |
|---|---|---|
| 1st mode (Fundamental) | n1 = \[\frac {v}{4L}\] | 1st harmonic |
| 2nd mode | n2 = \[\frac {3v}{4L}\] = 3n1 | 3rd harmonic or 1st overtone |
| pth | np = (2p − 1)\[\frac {v}{4L}\] | (2p − 1)th harmonic or (2p − 3)th overtone |
Formula: Open Organ Pipe
| Mode | Formula | Also Known As |
|---|---|---|
| 1st mode (Fundamental) | n1 = \[\frac {v}{2L}\] | 1st harmonic |
| 2nd mode | n2 = \[\frac {v}{L}\] = 2n1 | 2nd harmonic or 1st overtone |
| pth mode | np = p\[\frac {v}{2L}\] | pth harmonic or (p − 1)th overtone |
Formula: Wavelength & Length Relations
| Pipe | Length of pipe | Possible wavelengths |
|---|---|---|
| Closed | L = (n + \[\frac {1}{2}\])\[\frac {λ}{2}\], for n = 0,1,2,… | λ = \[\frac {2L}{(n+\frac {1}{2})}\] |
| Open | L = \[\frac {nλ}{2}\], where n=1,2,3,… | λ = \[\frac {2L}{n}\] |
Formula: Combined Law
Combining all three laws:
n = \[\frac {1}{2l}\]\[\sqrt {\frac {T}{m}}\]
where:
- n = fundamental frequency
- l = length of string
- T = tension in the string
- m = mass per unit length of the string
Formula: Beat Frequency
The number of beats formed per second is expressed as ∣v1 − v2∣, i.e., either (v1 − v2) or (v2 − v1), where v1 and v2 are frequencies of the two sound waves.
N = n1 − n2
Formula: Beat Period
The beat period is the reciprocal of beat frequency:
T = \[\frac{1}{n_1-n_2}\] or T = \[\frac{1}{|v_1-v_2|}\]
Formula: Fundamental Frequency of a Vibrating String
n = \[\frac{1}{2l}\sqrt{\frac{T}{\mu}}\]
where:
- l = length of string
- T = tension
- μ = mass per unit length (linear density)
Formula: Frequency of Beats
N = n1 - n2
Theorems and Laws [7]
Law: Principle of Superposition of Waves
When two or more waves arrive at a point simultaneously, each wave produces its own displacement independent of the other. The resultant displacement at a point is the vector sum (algebraic sum) of the displacements due to the individual waves.
State law of length.
The fundamental frequency of vibrations of a string is inversely proportional to the length of the vibrating string if tension and mass per unit length are constant.
∴ n ∝ `1/l` ...(if T and m are constant.)
Prove that the frequency of beats is equal to the difference between the frequencies of the two sound notes giving rise to beats.
Consider two sound waves, having the same amplitude and slightly different frequencies n1 and n2. Let us assume that they arrive in phase at some point x of the medium. The displacement due to each wave at any instant of time at that point is given as
`y_1 = A sin {2pi (n_1t - x/lambda_1)}`
`y_2 = A sin {2pi (n_2t - x/lambda_2)}`
Let us assume for simplicity that the listener is at x = 0.
∴ y1 = A sin (2πn1t) ...(i)
and y2 = A sin (2πn2t) ...(ii)
According to the principle of superposition of waves,
y = y1 + y2
∴ y = A sin (2πn1t) + A sin (2πn2t)
By using formula,
sin C + sin D = 2 sin `((C + D)/2) cos ((C − D)/2)`
y = `A[2sin((2pin_1t + 2pi n_2t)/2 )] cos [((2pin_1t - 2pin_2t)/2)]`
y = `2A sin [2pi ((n_1 + n_2)/2)t] cos [2pi ((n_1 - n_2)/2)t]`
∴ y = `R sin [2pi ((n_1 + n_2)/2)t]`
y = R sin (2πnt) ...(iii)
Where,
R = `2A cos[(2pi(n_1 - n_2))/(2)t]` and n = `(n_1 + n_2)/2`
Equation (iii) is the equation of a progressive wave having frequency `((n_1 + n_2)/2)` and resultant amplitude R.
For waxing,
A = ± 2a
`therefore 2A cos [2pi((n_1 - n_2)/2)t] = +- 2A`
`therefore cos [2pi ((n_1 - n_2)/2)]t = +-( 2A)/(2A)`
`therefore cos [2pi ((n_1 - n_2)/2)]t = +- 1`
This is possible if
`2pi ((n_1 - n_2)/2)t = 0, pi, 2pi, 3pi, ....`
i.e. t = 0, `1/(n_1 - n_2), 2/(n_1 - n_2), 3/(n_1 - n_2), ...`
∴ Period of beat T = `[1/(n_1 - n_2) - 0]`
T = `1/(n_1 - n_2)`
∴ Frequency of beats n = `1/T`
n = n1 − n2
Thus, the frequency of beats is equal to the difference between the frequencies of the two sound notes giving rise to beats.
State law of linear density.
The fundamental frequency of vibrations of a string is inversely proportional to the square root of mass per unit length (linear density) if the tension and vibrating length of the string are constant.
∴ `n ∝ 1/sqrtm` ...(if T and l are constant.)
Law: Low of Linear Density
The fundamental frequency of vibrations of a string is inversely proportional to the square root of mass per unit length (linear density), if the tension and vibrating length of the string are constant.
n ∝ \[\frac{1}{\sqrt{m}}\], if T and l are constant.
Law: Law of Length
The fundamental frequency of vibrations of a string is inversely proportional to the length of the vibrating string, if tension and mass per unit length are constant.
n ∝ \[\frac {1}{l}\], if T and m are constant.
Law: Law of Tension
The fundamental frequency of vibrations of a string is directly proportional to the square root of tension, if vibrating length and mass per unit length are constant.
n ∝ \[\sqrt {T}\] , if l and m are constant.
Key Points
Key Points: Free vs Forced Vibrations
| Sr. No. | Free Vibrations | Forced Vibrations |
|---|---|---|
| i. | Produced when a body is disturbed from its equilibrium position and released | Produced by an external periodic force of any frequency |
| ii. | Force is required initially only; then body vibrates on its own | Continuous external periodic force is required; if stopped, vibrations also stop |
| iii. | Frequency depends on the natural frequency of the body | Frequency depends on the frequency of the external periodic force |
| iv. | Energy remains constant in absence of friction/air resistance; decreases due to damping forces | Energy of the body is maintained constant by the external periodic force |
| v. | Amplitude decreases with time | Amplitude is small but remains constant as long as external periodic force acts |
| vi. | Vibrations stop sooner or later depending on the damping force | Vibrations stop as soon as the external periodic force is stopped |
Key Points: Beats
- Beats are formed when two waves of same amplitude but slightly different frequencies superimpose.
- Waxing and waning are alternatively produced.
- The greater the difference in frequency between the two waves, the higher the beat frequency.
Key Points: Properties of Progressive Waves
- In a progressive wave, each particle of the medium vibrates about its mean position in simple harmonic motion.
- All particles have the same amplitude, frequency, and time period.
- The phase of vibration changes from particle to particle.
- Energy is transferred through the medium, but no matter is transferred.
- Particles have maximum velocity at the mean position and zero velocity at extreme positions.
- Wave velocity depends on the properties of the medium.
- Progressive waves are of two types:
Transverse (vibrations perpendicular to the direction of propagation)
Longitudinal (vibrations parallel to the direction of propagation).
Key Points: Verification of Laws of Vibrating String
- First law (Law of length):
For constant tension and linear density, frequency is inversely proportional to length:
n ∝ \[\frac {1}{l}\] - Second law (Law of tension):
For constant length and linear density, frequency is directly proportional to the square root of tension:
n ∝ \[\sqrt {T}\] - Third law (Law of linear density):
For constant length and tension, frequency is inversely proportional to the square root of mass per unit length:
n ∝ \[\frac {1}{\sqrt {m}}\]These three laws are verified using a sonometer.
Key Points: Superposition of Waves
- When two waves meet, the resultant displacement is the sum of their individual displacements.
- After crossing, waves continue with their original shape and speed.
- In-phase waves (ϕ = 0) produce maximum amplitude (constructive interference).
- Out-of-phase waves (ϕ = π) produce a minimum or zero amplitude (destructive interference).
- Intensity depends on amplitude², so it is maximum in constructive and minimum in destructive interference.
Key Points: Progressive and Standing Waves
- A progressive wave travels through the medium, whereas a stationary wave does not.
- In progressive waves, all particles have the same amplitude; in stationary waves, the amplitude varies.
- In progressive waves, particles cross the mean position at different times; in stationary waves, they cross together.
- In progressive waves, all particles move; in stationary waves, nodes remain at rest.
- Progressive waves transfer energy; stationary waves do not transfer energy.
Key Points: Vibrations of a Stretched String
- A stretched string fixed at both ends forms stationary waves with nodes at the ends.
- The fundamental mode has one loop, with wavelength λ = 2l.
- Frequencies of harmonics are integral multiples of the fundamental:
np = pn - Frequency increases with tension and decreases with length and linear density.
Key Points; Application of Beats
- Beats are used to tune musical instruments; when beat frequency becomes zero, the instruments are in unison (same frequency).
- Beats are used in Doppler RADAR and SONAR to determine the speed of moving objects like aeroplanes, vehicles, and submarines.
- Beats help in finding an unknown frequency by adjusting a known frequency source until beat frequency becomes zero.
Key Points: Properties of Stationary Waves
Key Points: Vibrations of Air Columns and End Correction
- End correction is the extra length added because the antinode forms slightly outside the open end.
e ≈ 0.3d - For a closed pipe: corrected length L = l + e; only odd harmonics are present.
- For an open pipe: corrected length L = l + 2e; all harmonics are present.
- Fundamental frequency:
Closed pipe → n = \[\frac {v}{4L}\]
Open pipe → n = \[\frac {v}{2L}\] - For the same length, the fundamental frequency of an open pipe is double that of a closed pipe.
Key Points: Sound and Musical Instruments
- Sound has three main characteristics: loudness, pitch, and quality (timbre).
- Loudness depends on intensity and amplitude; sound level is measured in decibels (dB).
β = 10log10\[\frac {I}{I_0}\] - Pitch depends on frequency; a higher frequency means a higher pitch.
- Quality (timbre) depends on the number and relative amplitudes of overtones; it helps distinguish between sounds of the same pitch and loudness.
- A musical sound is produced by regular, periodic vibrations; irregular vibrations produce noise.
- Musical instruments work by producing stationary waves in strings, air columns, or membranes.
- Musical instruments are classified into stringed, wind, and percussion instruments.
Important Questions [41]
- Derive an expression for the equation of stationary wave on a stretched string.
- The equation of a simple harmonic progressive wave travelling on a string is y = 8 sin (0.02x - 4t) cm. The speed of the wave is ______.
- Show that the distance between two successive nodes or antinodes is λ/2.
- Phase difference between a node and an adjacent antinode in a stationary wave is ______.
- A Sonometer Wire Vibrates with Three Nodes and Two Antinodes, the Corresponding Mode of Vibration is
- Distinguish Between Forced Vibrations and Resonance.
- Differentiate between free and forced vibrations.
- Show that Only Odd Harmonics Are Present in an Air Column Vibrating in a Pipe Closed at One End.
- What Are Forced Vibrations and Resonance? Show that Only Odd Harmonics Are Present in an Air Column Vibrating in a Pipe Closed at One End.
- A Stretched Wire Emits a Fundamental Note of Frequency 256 Hz. Keeping the Stretching Force Constant and Reducing the Length of Wire by 10 Cm, the Frequency Becomes 320 Hz. Calculate the Original Length of Wire.
- State law of length.
- What are harmonics?
- Distinguish between an overtone and harmonic.
- What are overtones?
- Doppler Effect is Not Applicable When
- Find the Velocity of Sound in the Air and Frequency of the Tuning Fork
- In Doppler Effect of Light, the Term “Red Shift” is Used for
- The Working of Radar is Based on
- Apparent Frequency of the Sound Heard by a Listener is Less than the Actual Frequency of Sound Emitted by Source. in this Case
- State Any Four Applications Of Doppler Effect
- Prove that the frequency of beats is equal to the difference between the frequencies of the two sound notes giving rise to beats.
- Two tuning forks of frequencies 320 Hz and 340 Hz are sounded together to produce a sound wave. The velocity of sound in air is 326.4 m/s. Calculate the difference in wavelengths of these waves.
- The string of a guitar is 80 cm long and has a fundamental frequency of 112 Hz. If a guitarist wishes to produce a frequency of 160 Hz, where should the person press the string?
- Two tuning forks having frequencies 320 Hz and 340 Hz are sounded together to produce sound waves. The velocity of sound in air is 340 m/s. Find the difference in wavelength of these waves.
- Show that Only Odd Harmonics Are Present as Overtones in the Case of an Air Column Vibrating in a Pipe Closed at One End.
- Show that Even as Well as Odd Harmonics Are Present as Overtones in the Case of an Air Column Vibrating in a Pipe Open at Both the Ends.
- A Wheel of Moment of Inertia 1 Kg.M2 is Rotating at a Speed of 30 Rad/S. Due to Friction on the Axis, It Comes to Rest in 10 Minutes. Calculate the Average Torque of the Friction.
- State the Cause of End Correction.
- Discuss Different Modes of Vibrations in an Air Column of a Pipe Open at Both the Ends.
- Two Sound Notes Have Wavelengths 83/170 M and 83/172 M in the Air. These Notes When Sounded Together Produce 8 Beats per Second. Calculate the Velocity of Sound in the Air And Frequencies of the Two Notes.
- A pipe open at both ends resonates to a frequency ‘n1’ and a pipe closed at one end resonates to a frequency ‘n2’. If they are joined to form a pipe closed at one end, then the fundamental frequency will be
- State law of linear density.
- A pipe which is open at both ends is 47 cm long and has an inner diameter 5 cm. If the speed of sound in air is 348 m/s, calculate the fundamental frequency of air column in that pipe.
- With a neat labelled diagram, show that all harmonics are present in an air column contained in a pipe open at both the ends. Define end correction.
- The Value of End Correction for an Open Organ Pipe of Radius 'R' is
- A Tube Open at Both Ends Has Length 47 Em. Calculate the Fundamental Frequency of Air Column. (Neglect End Correction. Speed of Sound in Air is 3.3 X 102m/S
- Draw Neat Labelled Diagrams for Modes of Vibration of an Air Column in a Pipe When It Is Closed at One End.
- In a set, 21 turning forks are arranged in a series of decreasing frequencies. Each tuning fork produces 4 beats per second with the preceding fork. If the first fork is an octave of the last fork, find the frequencies of the first and tenth fork.
- Find the End Correction for the Pipe Open at Both the Ends In Fundamental Mode.
- Calculate the Ratio of Lengths of Their Air Columns
- What is Meant by Hamonics?
