Wave on a string

Any way to calculate the wave speed? What is it likely to depend on?

Amplitude of the wave? Wave length? Mechanical properties of the string?

All of those options are plausible, but it turns out the wave speed only depends on mass of the string (rope) and its tension.

Wave on a string

Waves on a string resemble very much harmonic oscillations of a mass on a spring.

Tension provides the restoring force, which wants to make the oscillations more frequent.

Mass of the string provides the inertia, which slows down oscillations and wave propagation.

Speed of a wave on a string

• Speed of a wave on a stretched string under tension, F

F

v

= m/L is the mass per unit length of the string

The wave speed is found as

Fv

f1

The frequency of oscillations is

NgmF load 20

m6stringstring m

L

m

How does v change if we increase tension by a factor of 2? make the string twice thicker?

m

kf

21

cf. pendulum

Frequency of a wave on a guitar string

Fv

f1

Wavelength, = 2L, where L is the length of the string.

If the string has a mass m, = m/L, and we have

Lm

F

Lm

F

Lm

F

Lf

42

1

/2

1

Piano: thick, heavy, and long strings – low pitch.

Tight strings – high pitch.

Compare: a mass on a spring.

L

F

4Spring constant(restoring force)

vs.

m

kf

k

A propagating wave communicates motion and carries away energy.

We can define wave power as amount of energy carried away from the source (and transported through the medium) per unit time

Wave power: how much energy is transported per unit time.

Measured in Joules/sec, J/s or Watts W.

)(sin 22 tkxkAFP For a harmonic wave propagating along a string

If we average it over time, at any position, x, we find that

2/1)(sin 2 tkx Therefore the power averaged over time 2

2

1kAFP

vk /We can plug in and to get2vF

vAP 22

2

1

A = v0 is the amplitude of the velocity of oscillations of

the material of the string along the y-axis

is the kinetic energy of oscillations per unit length

The power is proportional to the mass per unit length, to the square of the amplitude of oscillations of the velocity of string material and to the wave speed.

Let’s take a closer look at the power equation. What kind of sense does it make? vvvAP 2

022

2

1

2

1

is the mass per unit length;

is the angular frequency of oscillations in the wave

A is the amplitude of the oscillations

v is the wave speed

202

1v

Wave front is a continuous line or a surface connecting nearby wave crests.

Wave fronts are flat surfaces for well directed light, radar or sound beams, which propagate in one direction without spreading

Wave fronts are straight lines for ripples on water surface at shore line.

Plane waves

Wave front is a continuous line or a surface connecting nearby wave crests.

Wave fronts are spherical surfaces for spherical waves originating from a point source and propagating in 3D space.

Wave fronts are circles for waves on water surface originating from a point source.

Spherical and circular waves becomes less intense as they travel further away from their source, because the same power emitted by the source is spread over larger area (circumference).

A

PI

Wave intensity is the wave power per unit area

For a plane wave the intensity remains constant.For a spherical wave it

decreases with the distance, r, from the source, like

Measured in 22 m

Wor

ms

J

24 r

PI

Power P of the source is 1000 watts.The energy spreads outward through the air.  

At a distance R from the source the area

A of the spherical front is 4R2

Intensity = number of joules per second                   per square meter

               = power/area = P/A

Example: 1000 Joules per second of sound energy is generated. Spherical wave fronts spread outward from sound source at the center.

What is the intensity of sound at a distance of 10 m?

Is this a rock concert or a whisper?

m10R W1000P22 W/m8.0)4/(/ RPAPI

At the Earth surface the radius of curvature of the wave front is 150,000,000 km,The intensity is the same everywhere, and the wave is practically flat.

It is a spherical wave on the scale of the Solar system, though.

A light bulb is a source of a spherical wave.

If you are 5 ft away from the bulb and walk another 5 ft away the light intensity decreases by a factor of 22 = 4.

24 r

PI

In terms of intensity of sunlight, it is about the same as going from Venus to mars