Question

In a plane progressive harmonic wave, particle speed is always less than the wave speed, if:-
A. Amplitude of wave is less than $${\displaystyle \frac{\lambda }{2\pi }}$$
B. Amplitude of wave is greater than $${\displaystyle \frac{\lambda }{2\pi }}$$
C. Amplitude of wave is less than $$\lambda$$
D. Amplitude of wave is greater than $${\displaystyle \frac{\lambda }{\pi }}$$

Answer 1

Hint: The particle of the medium will oscillate along the vertical direction as the plane progressive harmonic wave travels. For the particle we use the sine wave function of displacement to find the velocity or displacement. The maximum velocity of the cork will be at the mean equilibrium position.

Formula used:
$$v=\nu \lambda$$
where v is the wave speed, $$\nu$$ is the frequency of the wave and $$\lambda$$ is the wavelength of the wave.
$$v_{max}=\omega A$$
where $$v_{max}$$ is the maximum speed of simple harmonic motion, $$\omega$$ is the angular frequency and A is the amplitude of the oscillation.

Complete step by step solution:
Let the particle speed is $$v_p$$and the wave speed is $$v_w$$. The general equation of the progressive wave is given as,
$$y\left(x,t\right)=A\sin \left(\omega t-kx\right)$$
Here, A is the amplitude of the vertical displacement, $$\omega$$ is the angular frequency of the wave and k is the wave number of the wave.

The wave speed is given as,
$$v_w={\displaystyle \frac{\omega }{K}}$$
Here, K is the wave number and it is related to the wavelength as,
$$K={\displaystyle \frac{2\pi }{\lambda }}\dots \left(i\right)$$
The particle speed is given as,
$$v_p=\omega \sqrt{A^2-y^2}$$

So, the maximum speed of the particle is when it is at mean position, i.e. the displacement is zero.
$$v_{p\left(max\right)}=\omega A$$
As per the question the particle speed is less than the wave speed,
$$v_{p\left(max\right)}<v_w$$
$$\Rightarrow \omega A<{\displaystyle \frac{\omega }{K}}$$
$$\Rightarrow A<{\displaystyle \frac{1}{K}}$$
From the first relation,
$$A<{\displaystyle \frac{\lambda }{2\pi }}$$
Hence, if the amplitude is less than $${\displaystyle \frac{\lambda }{2\pi }}$$ then the particle speed is always less than the wave speed.

Therefore, the correct option is A.

Note: We should be very clear that when a progressive wave is traveling through a medium then the particle of the medium oscillates about the mean position with varying speed which depends on the displacement from the mean position and the wave speed is constant.