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
