Energy transferred by string waves
u k  dK dx  12 μv y2  12 μy x,t  t   12 μv 2 y x,t  x 
2
uU  dU dx
2
v 2  F
u k  uU
 y x, t 
2  y  x, t  
u  u k  uU  v 
 F


 x 
 x 
2
2
u is maximum where the
slope is maximum
u is zero where the slope
is zero
For sinusoidal waves:
u  μ 2 A 2 sin kx  t 
2
u ev 
1
2
μ 2 A 2
E  u ev   vA 2
Pave

1
2
E 1

 2 v 2 A 2 
T
F 2 2
2 2
F  A   A
2v
T
1
1
2
sin kx  t  

T0
2
Question
A stretched string has a linear mass
density  = 0.010 kg/m. A
sinusoidal traveling wave moving on
the string has the wave function
The average power transmitted by
the wave is __ W.
y = (0.010 m)cos[(1.0 rad/m)x
 (100 rad/s)t]
Recall: v = /k.
1.
2.
3.
4.
0.5
1.0
1.5
2.0
Interference
Interference – combination of waves
(an interaction of two or more waves arriving at the same place)




(
r
,
t
)


(
r
,
t
)


(
r
, t)
Important: principle of superposition
1
2
Valley
Peak
(b)
(a)
Waves source
(b)
(a)
Valley
No shift or shift by
r2  r1   m
Shift by
r2  r1  m  12 
m  0,1,2,...
(a) If the interfering waves add up so that they reinforce each other, the total
wave is larger; this is called “constructive interference”.
(b) If the interfering waves add up so that they cancel each other, the total
wave is smaller (or even zero); this is called “destructive interference”.
Standing waves on a string
Wave interference, boundaries, and superposition
– Waves in motion
from one boundary
(the source) to
another boundary
(the endpoint) will
travel and reflect.
As wave pulses travel,
reflect, travel back,
and repeat the whole
cycle again, waves in
phase will add and waves
out of phase will cancel.
Standing waves on a string
1 
L  1
1
2
L  22 2  2
L  32 3
L  n2 n
2L
2 
L
2
2L
3 
3
n 
v
2L
 2L
1
2L
n
v
fn 
n
 nf1
n
2L
Different boundary conditions:
•Both ends fixed (see above)
•Both ends free (the same as above)
•One end fixed and on end free (next slide)
n=1,2,3...
f n  nf1
Standing waves on a string
One end fixed and on end free
n=1
L  14 1
n=3
L  34 3
L  n4 n
v
n
Tn
 n f n
v
vn
fn 

 f1 n
n 4 L
4L
n 
n
n  1,3,5...
Principle of superposition and standing waves
y1 ( x, t )  A cos( kx  t )
y 2 ( x, t )   A cos( kx  t )
y( x, t )  y1 ( x, t )  y2 ( x, t )  A cos(kx  t )  A cos(kx  t )
y( x, t )  2 A sin( kx)sin( t )   Asw sin( kx)sin( t )
Asw  2 A
Question
A stretched string between 2 fixed
ends has:
Length L = 1.0 m
Wave speed v = 100 m/s.
The fourth harmonic frequency of
vibration of the string is ___ Hz.
1. 50
2. 100
3. 150
4. 200
Question
A stretched string has one free end
and one fixed end, and is vibrating
at its 5th harmonic frequency.
The number of nodes in the wave
function is ___.
1.
2.
3.
4.
1
2
3
4