Question

To understand the standard formula for a sinusoidal traveling wave. One formula for a wave with a y displacement (e.g., of a string) traveling in the x direction is y(x,t)=Asin(kx−ωt).
All the questions in this problem refer to this formula and to the wave it describes.
1) What is the phase ϕ(x,t) of the wave? Express the phase in terms of one or more given variables ( A, k, x, t, and ω) and any needed constants like π
ϕ(x,t)=
2) What is the wavelength λ of the wave? Express the wavelength in terms of one or more given variables ( A, k, x, t, and ω) and any needed constants like π.
λ=
3) What is the period T of this wave? Express the period in terms of one or more given variables ( A, k, x, t, and ω) and any needed constants like π.
T=
4) What is the speed of propagation v of this wave? Express the speed of propagation in terms of one or more given variables ( A, k, x, t, and ω) and any needed constants like π.
v=

Answer 1

Explanation:

The displacement of the string in the x direction is given by :

$y(x,t)=A\ sin(kx-\omega t)$

Where

A is the amplitude of wave

k is the propagation constant

1. Here, the phase of the wave is $(kx-\omega t)$

2. The propagation constant of a wave is given by :

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

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

3. Since, $\omega=2\pi f$

$f=\dfrac{\omega}{2\pi}$

Time period, $T=\dfrac{1}{f}$

$T=\dfrac{2\pi}{\omega}$

4. Speed of this wave is given by :

$v=f\times \lambda$

$v=\dfrac{\omega}{2\pi}\times \dfrac{2\pi}{k}$

$v=\dfrac{\omega}{k}$

Hence, this is the required solution.