Superposition of Waves

Superposition of Waves Identical waves in opposite directions: “standing waves” 2 waves at slightly different frequencies: “beats” 2 identical waves, but not in phase: “interference”

Principle of Superposition 2 Waves In The Same Medium: The observed displacement y(x,t) is the algebraic sum of the individual displacements: y(x,t) =y1(x,t) + y2(x,t) (for a “linear medium”)

What’s Special about Harmonic Waves? 2 waves, of the same amplitude, same angular frequency and wave number (and therefore same wavelength) traveling in the same direction in a medium but are out of phase: Trig: sin a + sin b = 2 cos [(a-b)/2] sin [(a+b)/2] Result: Resultant amplitude

Assume  is positive: For what phase difference  is the total amplitude 2A? For what phase difference  is the total amplitude A? For what phase difference  is the total amplitude 0? For what phase difference  is the total amplitude A/2?

wave 1 wave 2 Resultant: Sine wave, same f, different A, intermediate f

Standing Waves Two identical waves traveling toward each other in the same medium with the same wave velocity y1 = Aosin(kx – ωt) y2 = Aosin(kx + ωt) Total displacement, y(x,t) = y1 + y2

Trigonometry : Then: This is not a traveling wave! Instead this looks like simple harmonic motion (SHM) with amplitude depending on position x along the string.

For fixed x,the particle motions are simple harmonic oscillations: t = 0, T, 2T…. y x -2A0 t = T/2, 3T/2, … node node node (no vibration)

kL=n π , n=1,2,3… determines allowed wavelengths Nodes are positions where the amplitude is zero. In particular there must be a node at each end of the string when the string is fixed at each end. kL=n π , n=1,2,3… determines allowed wavelengths Since it follows that 2L=n n , for n=1,2,3….

The n-th harmonic has n antinodes Nodes are ½ wavelength apart. Antinodes (maximum amplitude) are halfway between nodes.

Since: and , is the fundamental frequency The allowed frequencies on a string fixed at each end are

Practical Setup: Fix the ends, use reflections. We can think of traveling waves reflecting back and forth from the boundaries, and creating a standing wave. The resulting standing wave must have a node at each fixed end. Only certain wavelengths can meet this condition, so only certain particular frequencies of standing wave will be possible. example: (“fundamental mode” n=1) or first harmonic node node L

λ2 Second Harmonic λ3 Third Harmonic . . . .

In this case (a one-dimensional wave, on a string with both ends fixed) the possible standing-wave frequencies are multiples of the fundamental: f1, 2f1, 3f2, etc. This pattern of frequencies depends on the given boundary conditions. The frequencies satisfy a different rule when one end of the string is free.

Problem wave at t=0 and T y x 1.2 m f = 150 Hz 8mm x 1.2 m f = 150 Hz Write out y(x,t) for the standing wave.

Question m When the mass m is doubled, what happens to a) the wavelength, and b) the frequency of the fundamental standing-wave mode?

Example 120 cm Oscillator drives Standing waves with Constant frequency. m If the frequency of the standing wave remains constant what happens when the weight (tension) is quadrupled?