PHY 102: Waves & Quanta
Topic 3
Energy in wave motion
John Cockburn (j.cockburn@... Room E15)
•Power transmitted in a wave
•Sinusoidal wave on a string
•Wave Intensity
•Inverse Square Law
•The decibel scale
Example Calculation
One end of a 10m-long rope is attached to a post, and
pulled taut, to a tension of 140N. If the mass of the rope is
800g, what is the speed of transverse waves on the rope?
If the free end of the rope is waggled up and down with a
frequency of 1.2Hz, what is the wavelength of the
transverse waves on the rope?
Energy in Wave Motion
Waves transport energy. Examples??
How can we quantify this??
Wave on a string
T
A
Ty
y
Consider force in y-direction at point A on string:
x
y

T
x
Ty
Wave on a string
If a force F acts to move an object with velocity v:
Rate of energy transfer (power) = F• v =Fv for F//v
So for our wave on a string, P = Tyv……………..
ie
y ( x, t ) y ( x, t )
P ( x, t )  T
x
t
Consider case for sinusoidal wave………………
P( x, t )  T  A sin (t  kx)
2
2
2
Sinusoidal Wave on a string
Pmax  T  A
2
Pave
2
1

T  2 A 2
2
NB, for all MECHANICAL waves, Pave  A22
Example Calculation
A piano wire with mass 3g and length 80cm is stretched
with a tension of 25N. A wave with frequency 120Hz
and amplitude 1.6mm travels along the wire.
a) Calculate the average power carried by the wave
b) What happens to the average power if the wave
amplitude is halved?
Wave Intensity
Intensity I :
Average rate at which energy is transported by
the wave through unit surface area
perpendicular to direction of propagation
(average power per unit area)
For waves spreading out equally in three dimensions from a
point source, the power at distance r is distributed evenly
over a sphere of radius r, surface area 4r2....................
Inverse square law
Pave
I
4r 2
4r1 I 1  4r2 I 2 (assuming no energy absorption )
2
I 1 r22
 2
I 2 r1
2
Example Calculation
(a) For the piano in the previous question, calculate the
sound intensity at a distance of 5m
(b) How much does the sound intensity decrease if you
move to a distance of 15m from the piano?
The decibel scale
Used to provide a logarithmic scale for comparison of 2 quantities.
Often (but not only) used for sound intensities:
 (dB)  10 log 10
I1
I0
Examples
a) What is the attenuation in dB of the sound intensity
heard by a listener when they move from an initial
distance of 3m to a final distance of 24m from the
source?
b) An electronic amplifier increases the amplitude of
an AC signal from 10mV to 5V. What is the gain in
dB of the amplifier?
Sound Intensities
The dB scale is often used as an “absolute” measure of sound
intensity (loudness)
(eg the average sound intensity for traffic on the M1 is 75dB, or
whatever)
Here, the intensity is measured relative to the threshold of
human hearing:
I0 = 10-12 W/m2