MECHANICAL WAVES.
QUESTIONS
(1) Why do you see lightning before you hear the thunder? A familiar rule of thumb is to start
counting slowly, once per second, when you see the lightning; when you hear the thunder,
divide the number you have reached by 3 to obtain your distance from the lightning in
kilometers (or divide by 5 to obtain your distance in miles). Why does this work, or does
it?
(2) Children make toy telephones by sticking each end of a long string through a hole in the
bottom of a paper cup and knotting it so it will not pull out. When the spring is pulled
taut, sound can be transmitted from one cup to the other. How does this work? Why is the
transmitted sound louder than the sound traveling through air for the same distance?
(3) A musical interval of an octave corresponds to a factor of 2 in frequency. By what factor
must the tension in a guitar or violin string be increased to raise its pitch one octave? To
raise it two octaves? Explain your reasoning. Is there any danger in attempting these
changes in pitch?
PROBLEM 1. Mechanical waves. Mathematical description
A certain transverse wave is described by:
t
 x

y ( x, t )  (6.50mm) cos 2 


 28.0cm 0.0360 s 
Determine the wave’s (a) Amplitude; (b) wavelength (c) frequency (d) speed of propagation (e)
direction of propagation.
PROBLEM 2. Speed of Propagation vs. Particle Speed.
QUESTION
For transverse waves on a string, is the wave speed the same as the speed of any part of the
string? Explain the difference between these two speeds. Which one is constant?
PROBLEM
(a)
Show that the wave equation: y ( x, t )  A cos( kx  t ) may be written as:
 2

y ( x, t )  A cos  ( x  vt )  where v is the wave speed of propagation


(b)
U
Use y(x,t) too find an exppression forr the transveerse velocityy v y ( x, t ) off a particle iin the
(c)
sstring on whhich the wavve travels.
F
Find the maaximum speeed of the parrticle in the string. Undder what cirrcumstancess is this
eequal to the propagatioon speed v? Less than v? Greater thhan v?
Answ
wer:
PRO
OBLEM 3. Wave equaation is a lin
near equatiion.
Let y1 ( x, t )  A cos(k
e solutions oof the wave equation
c
1 x  1t ) and y2 ( x , t )  A cos( k 2 x   2 t ) be
for the
t same v. Show
S
that y ( x, t )  y1 ( x, t )  y2 ( x, t ) is also a solution of the wave eqquation.
Answ
wer:
The wave equattion is: