Sample Homework Problem
(a)
Write an equation describing a sinusoidal transverse wave traveling on a cord in
+x direction with a wavelength of 10 cm, a frequency of 400 Hz, and an
amplitude of 2.0 cm.
The general wave equation is: y ( x ,t ) = A sin( kx − ωt )
A = 2.0 cm
k = 2π/λ = 2π/10 cm-1 = 0.628 cm-1
ω = 2πf = 2π(400 s-1) = 2510 s-1
y(x,t) = (2.0 cm) sin (0.628 cm-1 x – 2510 s-1 t) or
y(x,t) = (2.0 cm) sin (62.8 m-1 x – 2510 s-1 t)
(b)
What is the maximum speed of a point on the cord?
This question is not asking for the wave speed. It is asking what the maximum
speed a point of the string has in its up-and-down motion. We have an expression
for the y displacement as a function time, t, in part (a). The up-and-down speed is
 ∂y( x ,t ) 
 ∂y( x ,t ) 
at a point is just v y = 
 . We want the maximum of v y = 
 .
 ∂t  x
 ∂t  x
Using the expression in (a) for y(x,t), we have:
 ∂y( x ,t ) 
-1
-1
-1
vy = 
 = (2.0 cm)(2510 s ) cos (0.628 cm x – 2510 s t)
∂
t

x
The maximum occurs when cosine term is 1.0.
Thus, maximum is vy = (2.0 cm)(2510 s-1) = 5020 cm/s ≈ 50 m/s.
(c)
What is the speed of the wave?
We can use v =
ω
k
=
2510
cm/s = 4000 cm/s = 40 m/s.
0.628