Question

Given that y is the vertical displacement of a wave at anytime and A is the amplitude of the wave, w is the angular speed of the wave and k is the wave number. If x is the horizontal distance covered by the wave at anytime t, show that for a progressive wave
Y = A Sin (wt + kx).

Answer 1

Explanation:

The general equation describing a wave is:

y(x,t) = A sin(kx - wt)

Let's say that for a particular wave on a string the equation is:

y(x,t) = (0.9 cm) sin[(1.2 m-1)x - (5.0 s-1)t]

(a) Determine the wave's amplitude, wavelength, and frequency.

(b) Determine the speed of the wave.

(c) If the string has a mass/unit length of m = 0.012 kg/m, determine the tension in the string.

(d) Determine the direction of propagation of the wave.

(e) Determine the maximum transverse speed of the string.

Solutions

Part (a): The wave's amplitude, wavelength, and frequency can be determined from the equation of the wave:

y(x,t) = (0.9 cm) sin[(1.2 m-1)x - (5.0 s-1)t]

The amplitude is whatever is multiplying the sine.

A = 0.9 cm

The wavenumber k is whatever is multiplying the x:

k = 1.2 m-1The wavelength isl=2pk= 5.2 m

The angular frequency w is whatever is multiplying the t.

w = 5.0 rad/sf=w2p= 0.80 Hz

Part (b): The wave speed can be found from the frequency and wavelength:

v = f l = 0.80 * 5.2 = 4.17 m/s

Part (c): With m = 0.012 kg/m and the wave speed given by:v=(Tm)½

This gives a tension of T = m v2 = 0.012 (4.17)2 = 0.21 N.

Part (d): To find the direction of propogation of the wave, just look at the sign between the x and t terms in the equation. In our case we have a minus sign:

y(x,t) = (0.9 cm) sin[(1.2 m-1)x - (5.0 s-1)t]

A negative sign means the wave is traveling in the +x direction.

A positive sign means the wave is traveling in the -x direction.

Part (e): To determine the maximum transverse speed of the string, remember that all parts of the string are experiencing simple harmonic motion. We showed that in SHM the maximum speed is:

vmax = Aw

In this case we have A = 0.9 cm and w = 5.0 rad/s, so:

vmax = 0.9 * 5.0 = 4.5 cm/s

maybe this should help