Question

A sinusoidal sound wave moves through a medium and is described by the displacement wave function

s(x, t) = 2.00 cos (15.7x ? 858t)

where s is in micrometers, x is in meters, and t is in seconds. Find (a) the amplitude, (b) the wavelength, and (c) the speed of this wave.
Determine the instantaneous displacement from equilibrium of the elements of the medium at the position x

Answer 1

Answer

given,

displacement wave function

s(x, t) = 2.00 cos (15.7 x + 858 t)

now, comparing the wave equation with general equation.

s(x, t) = A cos (k x + ω t)

where A is the amplitude of the wave in micrometer.

now,

a) Amplitude of the wave

   A = 2 x 10⁻⁶ μ m

b) $\lambda = \dfrac{2\pi}{k}$

       k = 15.7 m

   $\lambda = \dfrac{2\pi}{15.7}$

             λ = 0.4 m

c) wave speed

    $v = \dfrac{\omega}{k}$

    $v = \dfrac{858}{15.7}$

           v = 54.65 m/s

d) For instantaneous displacement

Assuming the position and time is given as

x = 0.05 m and t = 3 m s

now,

s(x, t) = 2.00 cos (15.7 x 0.05 + 858 x 3 x 10⁻³ )

s(x, t) = 2.00 cos (3.359)

s(x,t) = -1.95 μ m