1. Physics 1B03summer-Lecture 9
Wave Motion
Energy and power in sinusoidal waves
Physics 1B03summer-Lecture 9

2. Physics 1B03summer-Lecture 9
Energy in Waves
as waves propagate through a medium, they transport energy eg: ship moving up and down on a lake eg: feeling sound waves at a rock concert
hence, we can talk about energy and the ‘rate of energy transfer’
Physics 1B03summer-Lecture 9

3. Energy and Power dm dm = μ dx (μ = mass/unit length)
A stretched rope has energy/unit length: ds dm dx
For small A and large l, we can ignore the difference between “ds”, “dx” :
dm = μ dx (μ = mass/unit length)
Physics 1B03summer-Lecture 9

4. Physics 1B03summer-Lecture 9
The mass dm vibrates in simple harmonic motion. Its maximum kinetic energy is
dKmax = ½(dm)vmax2
= ½(dm)(ωA)2
The average kinetic energy is half this maximum value, but there is also an equal amount of potential energy in the wave. The total energy (kinetic plus potential) is therefore:
dE = ½(dm) ω 2A2
To get the energy per unit length (or energy ‘density’), replace the mass dm with the mass per unit length m:
Physics 1B03summer-Lecture 9

5. Physics 1B03summer-Lecture 9
Power: Energy travels at the wave speed v, So
waves on a string,
Both the energy density and the power transmitted are proportional to the square of the amplitude. This is a general property of sinusoidal waves.
Physics 1B03summer-Lecture 9

6. Physics 1B03summer-Lecture 9
Example
A string for which μ=5.0x10-2 kg/m is under tension of 80.0 N. How much power must be supplied to the string to generate sinusoidal waves at a frequency of 60Hz and with an amplitude of 6.0 cm ?
Physics 1B03summer-Lecture 9

7. Physics 1B03summer-Lecture 9
Example
A sinusoidal wave on a string is described by the equation: y(x,t) = (0.15m)sin(0.80x-50t)
where x is in meters and t in seconds. If μ=12.0g/m, determine:
the speed of the wave
the speed of particles on the wave at any time
the wavelength
the frequency
the power transmitted to the wave
Physics 1B03summer-Lecture 9

8. Physics 1B03summer-Lecture 9
Concept Quiz
The sound waves from your 100-watt stereo causes windows across the street to vibrate with an amplitude of 1 mm. If you use a 400-watt amplifier, what sort of amplitude can you get from the windows?
2mm
4mm
16 mm
Physics 1B03summer-Lecture 9

9. Physics 1B03summer-Lecture 9
Intensity
I = Power per unit area
Unit: W / m2
(the area is measured perpendicular to the wave velocity)
Intensity ~ (amplitude)2
source
detectors (area A)
Physics 1B03summer-Lecture 9

10. Physics 1B03summer-Lecture 9
How would the intensity depend on distance from the source for:
waves spreading out equally in all directions in space? (This is called an“isotropic” source, or a source of “spherical waves”.)
Waves spreading out on a two-dimensional surface, e.g., circular ripples from a stone dropped into water?
How would the amplitude depend on distance?
Physics 1B03summer-Lecture 9