Question

In the given progressive wave y = 5 sin(100πt − 0.4πx) Where y and x are in m, t is in s. What is the 

(a) Amplitude 

(b) Wavelength 

(c) Frequency

(d) Wave velocity

(e) Particle velocity amplitude.

Answer 1

Standard progressive wave-

y = asin(wt − kx + ϕ)

y = 5 sin(100πt − 0.4πx + 0)

(a) Amplitude, a = 5 m

(b) Wavelength, λ

k = \(\frac{2\pi}{\lambda}\) or  λ = \(\frac{2\pi}{k}\) 

k = 0.4π

λ = \(\frac{2\pi}{0.4\pi}\) = 5 m

(c) Frequency v,

w = 2πv

Or v = \(\frac{w}{2\pi}\) 

∵ w = 100π

v = \(\frac{100\pi}{2\pi}\) = 50 Hz

(d) Wave velocity, v = vλ = 50 × 5

= 250 \(\frac{m}{s}\) 

y = 5 sin(100πt − 0.4πx)

\(\frac{dy}{dx}\) = 5 × 100π cos(100πt− 0.4πx)

For maximum velocity at mean position –

cos(100πt − 0.4πx) = 1

Or 100πt − 0.4πx = 0

∴ \({(\frac{dy}{dx})}_{maximum}\)= 5 × 100π × 1

v_max = 500π m/s.