Physics Waves

Frequency of the whistle,  $\nu$ = 400 Hz

Speed of the train,  $v_r$ = 10 m/s

Speed of sound, v = 340 m/s

The apparent frequency (f') of the whistle as the train approaches the platform is given by the relation:

f' = ( $\frac{v}{v-v_r})\nu$ = ( $\frac{340}{340-10})*400=$ 412.12 Hz

The apparent frequency (f”) of the whistle as the train recedes from the platform is given by the relation:

f” = ( $\frac{v}{v+v_r})\nu$ = ( $\frac{340}{340+10})*400=$ 388.57 Hz

The apparent change in the frequency of sound is caused by the relative motions of the source and the observer. These relative motions produce no effect on the speed of sound.

Therefore, the speed of sound in air in both the cases remains the same, i.e. 340 m/s.

(a) A node is a point where the amplitude of vibration is the minimum and the pressure is maximum. On the other hand, an antinode is a point where the amplitude of vibration is maximum and the pressure is minimum. Therefore, a displacement node is nothing but a pressure antinode and vice versa.

(b) Bats emit very high-frequency ultrasonic sound waves. These waves get reflected back toward them by obstacles. A bat receives a reflected wave (frequency) and estimates the distance, direction, nature and size of the obstacle with the help of its brain senses.

(c) The overtone produced by a sitar and a violin, and the strengths of these overtones, are different. Hence, one can distinguish between the notes produced by a sitar and a violin even if they have the same frequency of vibration.

(d) Solids have shear modulus. They can sustain shearing stresses. Since fluids do not have any definite shape, they yield to shearing stress. The propagation of a transverse waves is such that it produces shearing stress in a medium. The propagation of such wave is possible only in solids, and not in gases. Both solids and fluids have their respective bulk moduli. They can sustain compressive stress. Hence, longitudinal waves can propagate through solids and fluids.

(e) A pulse is actually being a combination of waves having different wavelengths. These waves travel in a dispersive medium with different velocities, depending on the nature of the medium. This results in the distortion of the shape of a wave pulse.

(a) The general equation representing a stationary wave is given by the displacement function y(x,t) = 2asin kx cos $\omega t$

The equation is similar to the give equation y(x, t) = 0.06 sin ( $\frac{2\pi }{3}x)$ cos (120 $\pi t)$

Hence, the given function represents a stationary wave.

A wave travelling along the positive x-direction is given as: $y_1$ = asin ( $\omega$ t – kx)

The wave travelling along the negative x-direction is given as: $y_2$ = asin ( $\omega$ t + kx)

(b) The superposition of these two waves yields:

y = $y_1+y_2$ = asin ( $\omega$ t – kx) + asin ( $\omega$ t + kx)

= a sin( $\omega$ t)cos(kx) – a sin(kx)cos( $\omega$ t) – asin( $\omega$ t)cos(kx) – asin(kx)cos( $\omega$ t)

= -2a sin (kx) cos ( $\omega$ t)

= -2asin( $\frac{2\pi }{\lambda }$ x) cos (2 $\pi \nu$ t) …….(i)

The transverse displacement of the string is given as

y(x, t) = 0.06 sin ( $\frac{2\pi }{3}x)$ cos (120 $\pi t)\dots \dots \dots$ (ii)

Comparing equations (i) and (ii), we have $\frac{2\pi }{\lambda }$ = $\frac{2\pi }{3}$

Therefore $\lambda$ = 3m, it is given that 120 $\pi$ = 2 $\pi \nu$ so Frequency, $\nu =60\mathrm{H}\mathrm{z}$

Wave speed, v = $\nu \lambda$ = 60 $*3$ = 180 m/s

The velocity of a transverse wave travelling in a string is given by the relation:

v = $\sqrt{\frac{T}{\mu }}$ ….(i)

We have v = 180 m/s, mass of the string, m = 3 $*{10}^{-2}$ kg,

length of the string, l = 1.5 m

Mass per unit length of the string, $\mu$ = m/l = $\frac{3}{1.5}*{10}^{-2}$ = $2*{10}^{-2}$ kg/m

(c) Tension in the string, T = $v^2$ $*\mu ={180}^2$ $*2*{10}^{-2}$ = 648 N