Advertisements
Advertisements
प्रश्न
A wave propagates on a string in the positive x-direction at a velocity \[\nu\] \[t = t_0\] is given by \[g\left( x, t_0 \right) = A \sin \left( x/a \right)\]. Write the wave equation for a general time t.
Advertisements
उत्तर
Given,
Wave velocity = \[\nu\]
Shape of the string at
\[t = t_0\]
\[g\left( x, t_0 \right) = A \sin \left( x/a \right)\]
For a wave travelling in the positive x-direction, the general equation is given by \[y = A \sin \left( \frac{x}{a} - \frac{t}{T} \right)\]
Putting t = − t and comparing with equation (i), we get:
\[g\left( x, 0 \right) = A\sin\left\{ \left( \frac{x}{a} \right) + \left( \frac{t_0}{T} \right) \right\}\]
\[ \Rightarrow g\left( x, t \right) = A\sin\left[ \left\{ \left( \frac{x}{a} \right) + \frac{t_0}{T} \right\} - \left( \frac{t}{T} \right) \right]\]
\[Now, \]
\[T = \frac{a}{\nu}\]
\[Here, \]
a = Wave length
nu = Velocity of the wave
Thus, we have:
\[y = A\sin \left[ \left( \frac{x}{a} \right) + \frac{t_0}{\left( \frac{a}{\nu} \right)} - \frac{t}{\left( \frac{a}{\nu} \right)} \right]\]
\[\Rightarrow y = A\sin \frac{x + \nu \left( t_0 - t \right)}{a}\]
APPEARS IN
संबंधित प्रश्न
A stone dropped from the top of a tower of height 300 m high splashes into the water of a pond near the base of the tower. When is the splash heard at the top given that the speed of sound in air is 340 m s–1? (g= 9.8 m s–2)
Use the formula `v = sqrt((gamma P)/rho)` to explain why the speed of sound in air increases with humidity.
A bat emits an ultrasonic sound of frequency 1000 kHz in the air. If the sound meets a water surface, what is the wavelength of the transmitted sound? The speed of sound in air is 340 m s–1 and in water 1486 m s–1.
For the travelling harmonic wave
y (x, t) = 2.0 cos 2π (10t – 0.0080x + 0.35)
Where x and y are in cm and t in s. Calculate the phase difference between oscillatory motion of two points separated by a distance of `(3λ)/4`.
(i) For the wave on a string described in Exercise 15.11, do all the points on the string oscillate with the same (a) frequency, (b) phase, (c) amplitude? Explain your answers. (ii) What is the amplitude of a point 0.375 m away from one end?
A steel rod 100 cm long is clamped at its middle. The fundamental frequency of longitudinal vibrations of the rod is given to be 2.53 kHz. What is the speed of sound in steel?
A train, standing in a station-yard, blows a whistle of frequency 400 Hz in still air. The wind starts blowing in the direction from the yard to the station with at a speed of 10 m s–1. What are the frequency, wavelength, and speed of sound for an observer standing on the station’s platform? Is the situation exactly identical to the case when the air is still and the observer runs towards the yard at a speed of 10 m s–1? The speed of sound in still air can be taken as 340 m s–1.
Show that for a wave travelling on a string
\[\frac{y_{max}}{\nu_{max}} = \frac{\nu_{max}}{\alpha_{max}},\]
where the symbols have usual meanings. Can we use componendo and dividendo taught in algebra to write
\[\frac{y_{max} + \nu_{max}}{\nu_{max} - \nu_{max}} = \frac{\nu_{max} + \alpha_{max}}{\nu_{max} - \alpha_{max}}?\]
A sine wave is travelling in a medium. The minimum distance between the two particles, always having same speed, is
Two waves of equal amplitude A, and equal frequency travel in the same direction in a medium. The amplitude of the resultant wave is
A pulse travelling on a string is represented by the function \[y = \frac{a^2}{\left( x - \nu t \right)^2 + a^2},\] where a = 5 mm and ν = 20 cm-1. Sketch the shape of the string at t = 0, 1 s and 2 s. Take x = 0 in the middle of the string.
A wave pulse is travelling on a string with a speed \[\nu\] towards the positive X-axis. The shape of the string at t = 0 is given by g(x) = Asin(x/a), where A and a are constants. (a) What are the dimensions of A and a ? (b) Write the equation of the wave for a general time t, if the wave speed is \[\nu\].
A wave travelling on a string at a speed of 10 m s−1 causes each particle of the string to oscillate with a time period of 20 ms. (a) What is the wavelength of the wave? (b) If the displacement of a particle of 1⋅5 mm at a certain instant, what will be the displacement of a particle 10 cm away from it at the same instant?
A string of length 20 cm and linear mass density 0⋅40 g cm−1 is fixed at both ends and is kept under a tension of 16 N. A wave pulse is produced at t = 0 near an ends as shown in the figure, which travels towards the other end. (a) When will the string have the shape shown in the figure again? (b) Sketch the shape of the string at a time half of that found in part (a).
Following figure shows two wave pulses at t = 0 travelling on a string in opposite directions with the same wave speed 50 cm s−1. Sketch the shape of the string at t = 4 ms, 6 ms, 8 ms, and 12 ms.
An organ pipe of length 0.4 m is open at both ends. The speed of sound in the air is 340 m/s. The fundamental frequency is ______
What is the interference of sound waves?
For the travelling harmonic wave
y (x, t) = 2.0 cos 2π (10t – 0.0080x + 0.35)
Where x and y are in cm and t in s. Calculate the phase difference between oscillatory motion of two points separated by a distance of 4 m.
An engine is approaching a cliff at a constant speed. When it is at a distance of 0.9 km from cliff it sounds a whistle. The echo of the sound is heard by the driver after 5 seconds. Velocity of sound in air is equal to 330 ms-1. The speed of the engine is ______ km/h.
