Properties of waves: 1.Transverse 2.Longitudinal Propagation of the wave depends on the medium Medium does not travel with the wave (not exact – tsunami) Waves have magnitude and direction

Wave equation

Periodic Waves Functions which are periodic in time or distance x,t=  2  3  4  Period: x or t = 2 

Space Time Period of sin(nt) Note: frequency f Angular velocity  Period of sin(kx) Note: wavelength k is called the “wave number”

Space + time With what velocity does the wave travel? Amplitude

How to calculate the velocity of waves? If mass per unit length is 0.01 kgm -1 Sound waves R – ideal gas const. (as in PV=nRT) T – temp in kelvin M – molar mass (for air = 29  kg.mol -1 )  - is 1.4 for air (reflects how air molecules absorb energy)

Phase

Angular representation and phase   /8

Other forms of periodic waves:

Equation of motion: For particles For waves? – we need a wave equation

Equation of wave x y

How does the wave vary as either time or position is changed? The wave-equation tells us the answer. 1.Fix time calculate how y(x,t) changes with x i.e. 2. Fix position calculate how y(x,t) changes with t i.e.

Wave-eqn (presentation and demonstration) and are called first order partial derivatives Second order derivatives

Wave-equation for a string Assume that the string is under tension T (constant throughout) and the mass per unit length is  x y T T  ’’ ll xx A snapshot! The transverse force (in y-direction)

For small angles xx yy  Remember:

The mass of the section of string is: Apply F=ma – consider acceleration in y-direction Note before we had So as expected

What does it all mean? LHS is the curvature of the string (rate of change of grad.) small large RHS is the transverse acceleration of the string So acceleration is proportional curvature

Note that we calculate the curvature at a fixed time and the acceleration at a fixed position. So we should write

Reflection and transmission of waves

What happens when light and heavy strings are joined? Consider a travelling wave is incident from the left At the junction of the strings the displacement must be continuous 11 22 x=0

if not equal grad. then d 2 y/dx 2 is finite there is finite force on infinitesimal mass and thus inf. acceln!

The frequency with which the waves travel down the string is the same for both parts (depends on the frequency with which the waves are generated) So this is not really a solution! i.e. cannot satisfy the boundary conditions in this way.

Need to add to the transmitted wave in medium 2 a reflected wave in medium 1 x=0 incident reflected transmitted

 1  2 x=0  1  2

if the second string is massless then k 2 =0 and C=A if the second string has infinite mass then k 2 =  and C=-A What happens at the ends of the string?

What about light waves (electromagnetic)? To minimise reflections make n 1 =n 2

Standing Waves

Here we are applying the principle of superposition, i.e. we add the two individual displacements to get the total displacement. This is the equation of the stationary or standing waves

Spatial nodes when: Antinodes

Note: stationary waves are also a solution of the wave-equation

Stationary waves; normal modes x=0 x=L y What stationary waves can we have? Note change in phase of time part

frequency of the modes

Note: if the boundary conditions are different then we get a different solution for the standing waves. Not fixed Sound waves in a pipe/tube

Resonance tube (lab experiment) density pressure displacement

Waves in two dimensions Membrane Edge fixed Edge free Drum Beaker

Three dimensions

Energy in waves Travelling waves (on a string): , T y x dx dl kinetic energy of element, dx, of string strain energy (PE associated with stretching the string from dx to dl) dx dl if gradient is small formulae

extension of string: So the strain energy (PE) and kinetic energy are equal displacement KE/Strain E KE max KE min

PE max as string is stretched most here Intensity=P/Area So I  A 2

Energy in standing waves In standing waves nodes are fixed and thus E does not flow

Intensity and Loudness Our ears and eyes are logarithmic detectors…. thus a logarithmic scale has been adopted to measure intensity levels. I is the intensity of the source I 0 is is a reference level defined as W/m 2 (approx. threshold of hearing) Hearing threshold Pain threshold: I= 1 Wm -2

Superposition When waves interfere the result is the algebraic sum Interference of harmonic waves consider two waves formulae

 =0  =  /4  =  /2 ==

Beats Interference with sound waves with slightly different frequencies Lets choose a fixed point (say x=0) so kx is either a constant (or 0) in phase out of phase in phase out of phase

Amplitude oscillates with a frequency: Note: amplitudes may be different

“The timing of the pulses varies by less than one part in 10 15, making them more accurate than many atomic clocks. Gravity waves travelling through space between Earth and the pulsar should disrupt the precise timing of the pulses and thereby betray their presence to observers.” New Scientist Beats and Gravitational waves Next

Phase difference due to path difference suppose two sources S 1 and S 2 oscillate in phase and emit harmonic waves of the same frequency and wavelength. Now consider at the point P in space for which the path lengths differ. if there is an integral number of wavelengths, then interference is constructive, if half num. of wavelengths then it is destructive. P S1S1 S2S2

Coherence Two sources that are in phase or have a constant phase difference, are said to be coherent. Beams from independent sources are not coherent. Addition of coherent waves (constant phase difference – hence can add amplitudes) Intensity: Addition of incoherent waves (phase changes randomly – hence add intensities) Intensity:

Now put the space and time components together!

Doppler effect (for sound) If a wave source and receiver are moving relative to each other, the frequency detected by the receiver is not that emitted by the source Two scenarios (1) moving source (2) moving observer

1) moving source (receiver at rest) Source has frequency: f s (period T s =1/f s ) speed of source: u s velocity of waves in medium: v vT s usTsusTs b f

Case 2) receiver moving: If moving towards waves encounter more waves per second and thus higher frequency. Speed of receiver: u r urur Wavelength appears shorter Time between waves: T r

Doppler for light (relativity): Doppler shift in freq. depends on if source or receiver is moving relative to the medium. For light, for which there is no medium (propagates in vac), absolute motion cannot be detected; only relative motion of source and receiver can be determined. formulae

Shock waves: is what happens when the velocity of the source is faster than the velocity of waves in the medium

Shockwaves in light: Cerenkov radiation speed of particle in medium: u=  c speed of light in medium: v=c/n possible for speed of particle in medium to exceed speed of light