some general p roperties of waves

Some general properties of waves

Summing waves

• The wave equation is linear• A sum of waves will be the arithmetical sum of the

function representing them – and still a solution of the wave equation, but warning: energy is proportional to the square of the amplitude!

Beats

Group velocity and phase velocity

Standing (stationary) waves

Some very basic physics of stringed instruments……….

2f1

3f1

4f1

The fundamental frequency determines the pitch of the note.

the higher harmonics determine the “colour” or “timbre” of the note.

(ie why different instruments sound different)

Fundamental wavelength = 2L

From v = fλ,

f1= v/2L

So, for a string of fixed length, the pitch is determined by the wave velocity on the string…..

The string length on standard violin is 325mm. What tension is required to tune a steel “A” string (diameter =0.5mm) to correct pitch (f=440Hz)?

Density of steel = 8g cm

Changing the “harmonic content”

string plucked here

Plucking a string at a certain point produces a triangular waveform, that can be built up from the fundamental plus the higher harmonics in varying proportions.

Plucking the string in a different place (or even in a different way) gives a different waveform and therefore different contributions from higher harmonics (see Fourier analysis)

This makes the sound different, even though pitch is the same…………………

Doppler Effect

• The Doppler effect is the apparent change in the frequency of a wave motion when there is relative motion between the source of the waves and the observer.

• The apparent change in frequency f experienced as a result of the Doppler effect is known as the Doppler shift.

• The value of the Doppler shift increases as the relative velocity v between the source and the observer increases.

• The Doppler effect applies to all forms of waves.

Suppose the source moves at a steady velocity vs towards a stationary observer.

The source emits sound wave with frequency f.

From the diagram, we can see that the distance between crests is shortened such that sv'

vs

Since = c/f and = 1/f,

we get

fv

fc

fc s

'

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⇒ f ' = (c

c − vs) f = f

1

1 − vs c

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

Doppler Effect (moving source)

Consider an observer moving with velocity vo toward a stationary source S.

The source emits a sound wave with frequency f and wavelength = c/f.

The velocity of the sound wave relative to the observer is c + vo.

fcvcvcf oo

/'

fcvf o )1('

c

Doppler Effect (moving observer)

Doppler Shift

Consider a source moving towards an observer, the Doppler shift f is

€

f = f ' − f = f1

1 − vs c

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟− f

⇒ Δff

=1

1− vs c

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟−1

If vs<<c, then we get

€

ff

≅vsc

Δλλ

≅ −vsc

⎛ ⎝ ⎜

⎞ ⎠ ⎟

The above equation also applies to a receding source, with vs taking as negativeThe same equation applies for the moving observer (note the limit vs<<c)

• The velocities of distant galaxies can be determined from the Doppler shift ( The apparent change in frequency). • Light from such galaxies is shifted toward lower frequencies, indicating that the galaxies are moving away from us. This is called the red shift.

Applications of Doppler Effect (Astronomy)

Redshift

Blue shift

Red Shift

Hubble’s Law• Hubble found that (almost) every

galaxy was moving away from us.• Moreover, the further away it

was, the faster it was moving away from us.

This is a plot of some galaxies.

• The x axis is the distance to the galaxy• The y axis is the speed at which the galaxy is moving away from us

This line can be described by an equation which relates the distance to a galaxy to the recession velocity – Hubble's Law.

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v ≅ H0dH0 ≅ 25 km/s/Mly ≅1/(14 Gy)

What happens if vs > c

Cherenkov effect

Huygens’ Principle (conjectured in 1600)• All points on a given wave front can be taken

as point sources for the production of spherical secondary waves, called wavelets, which propagate in the forward direction with speeds characteristic of waves in that medium– After some time has elapsed, the new position of

the wave front is the surface tangent to the wavelets

• Demonstrated by Kirkhhoff in 1882, but Huygens was missing two points:– Amplitude varies as f(θ) ~ (1+cosθ)/2

– Phase is anticipated by π/2

In many problems these two points can be neglected

Huygen’s Construction for a Plane Wave

• At t = 0, the wave front is indicated by the plane AA’

• The points are representative sources for the wavelets

• After the wavelets have moved a distance cΔt, a new plane BB’ can be drawn tangent to the wavefronts

Huygen’s Construction for a Spherical Wave

• The inner arc represents part of the spherical wave

• The points are representative points where wavelets are propagated

• The new wavefront is tangent at each point to the wavelet

Huygen’s Principle and the Law of Reflection

• The Law of Reflection can be derived from Huygen’s Principle

• AA’ is a wave front of incident light

• The reflected wave front is CD

• Triangle ADC is congruent to triangle AA’C

• θ1 = θ1’• This is the Law of

Reflection

Huygen’s Principle and the Law of Refraction

• In time Δt, ray 1 moves from A to B and ray 2 moves from A’ to C

• From triangles AA’C and ACB, all the ratios in the Law of Refraction can be found– n1 sin θ1 = n2 sin θ2

Total Internal Reflection• Total internal reflection can occur when

light attempts to move from a medium with a high index of refraction to one with a lower index of refraction– Ray 5 shows internal reflection

• A particular angle of incidence will result in an angle of refraction of 90°– This angle of incidence is called the

critical angle

• For angles of incidence greater than the critical angle, the beam is entirely reflected at the boundary– This ray obeys the Law of Reflection at

the boundary

Optical fibers

• An application of internal reflection

• Plastic or glass rods are used to “pipe” light from one place to another

• Applications include– medical use of fiber optic

cables for diagnosis and correction of medical problems

– Telecommunications

Frequency Between Media

• As light travels from one medium to another, its frequency does not change– Both the wave speed and

the wavelength do change– The wavefronts do not pile

up, nor are created or destroyed at the boundary, so ƒ must stay the same

Fermat’s principle