Physics 1B03summer-Lecture 9

Wave MotionWave Motion

•Energy and power in sinusoidal waves

Physics 1B03summer-Lecture 9

Energy in Waves

- as waves propagate through a medium, they transport energy

eg: ship moving up and down on a lakeeg: feeling sound waves at a rock concert

- hence, we can talk about energy and the ‘rate of energy transfer’

Physics 1B03summer-Lecture 9

Energy and Power

)(amplitudePower Energy, 2

A stretched rope has energy/unit length:

dx

dsdm

For small A and large , we can ignore the difference between “ds”, “dx” :

dm = μ dx (μ = mass/unit length)

Physics 1B03summer-Lecture 9

The mass dm vibrates in simple harmonic motion. Its maximum kinetic energy is dKmax = ½(dm)vmax

2

= ½(dm)(ωA)2

dE = ½(dm) ω 2A2

22

21

length)(unit A

E

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:

To get the energy per unit length (or energy ‘density’), replace the mass dm with the mass per unit length :

Physics 1B03summer-Lecture 9

Power: Energy travels at the wave speed v,

So

waves on a string,

lengthEnergy

vP

vAP 2221

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

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

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:

a) the speed of the wave

b) the speed of particles on the wave at any time

c) the wavelength

d) the frequency

e) the power transmitted to the wave

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?

A) 2mmB) 4mmC) 16 mm

Physics 1B03summer-Lecture 9

IntensityIntensity

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

How would the intensity depend on distance from the source for:

1) waves spreading out equally in all directions in space? (This is called an“isotropic” source, or a source of “spherical waves”.)

2) 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?