Standing Wave & Resonance pattern that results when 2 waves, of same f, & A travel in opposite directions. Often formed from pulses reflect off a boundary. Waves interfere constructively (antinodes) & destructively (nodes) at fixed points.

Standing waves have no net transfer of energy – no direction propagation of energy.

Standing Waves form at natural frequencies of the material Standing Waves form at natural frequencies of the material. Occur when material resonates. When system is disturbed it vibrates at many frequencies. Standing wave patterns continue. Other frequencies to die out. Since there is resonance, the amplitude of particular wavelengths/frequencies will be amplified.

Relation of Wavelength to String Length for Standing Waves L = 1/2l.

l = L.

l= L

General expression relating wavelength to string length for standing waves: n ( ½ l) = L n is a whole number A whole number of half l’s must fit.

Although we would perceive a string vibrating as a whole, it vibrates in a pattern that appears erratic producing many different overtone pitches. What results are particular tone colors or timbres of instruments and voices.

Harmonics Each standing wave pattern= harmonic. The lowest f (longest l) at which a string can form standing wave pattern is the fundamental f or the first harmonic.

2nd Harmonic

Which One??

String Length L, l & Harmonics Standing waves can form on a string of length L, when the l can = ½ L, or 2/2 L, or 3/2L etc. Standing waves are the overtones or harmonics. L = nln. n = 1, 2, 3, 4 whole number harmonics. 2

Standing waves form where ½ l fits the string exactly, calculate f: Frequencies Standing waves form where ½ l fits the string exactly, calculate f: Substitute v/f for l. n = harmonic Must know speed in material.

1st standing wave forms when l = 2L First harmonic frequency is when n = 1 as below. When n = 1, f = v/l . This is fundamental frequency or 1st harmonic. First harmonic has largest amplitude.

For second harmonic n = 2. f2 = v/L Other standing waves with smaller wavelengths form other frequencies that ring out along with the fundamental. For second harmonic n = 2. f2 = v/L

In general, The harmonic frequencies can be found where n = 1,2,3… and n corresponds to the harmonic. v is the velocity of the wave on the string. L is the string length.

Pipes and Air Columns

closed-pipe resonator A resonant air column is simply a standing longitudinal wave system, much like standing waves on a string. closed-pipe resonator tube in which one end is open and the other end is closed open-pipe resonator tube in which both ends are open

Pipes – the open end has antinode (low P).

Standing Waves in Open Pipe Both ends must be antinodes Standing Waves in Open Pipe Both ends must be antinodes. How much of the wavelength is the fundamental?

The 1st harmonic or fundamental can fit ½ l into the tube. Just like the string L = nl 2 fn = nv 2L Where n, the harmonic is an integer.

Closed Pipe Resonator

Closed pipes must have a node at closed end and an antinode at the open end. How many wavelengths? L = l 4

Here is the next harmonic. How many l’s? L = 3l 4

There are only odd harmonics possible – n = odd number only. L = 1/4l. L = 3/4l. L = 5/4l. fn = nv where n = 1,3,5 … 4L

Application: When waves propagate through a tall building, the building resonates like a tube open at two ends. What is the equation that relates frequency to wave velocity and building height?

The building is 360 m tall and allows waves to travel through it at 2400 m/s, what frequency wave will cause the most damage to it? Explain why. (Hint: What is the resonant frequency)? 3.3 Hz

Hwk Read Homer section 4.5. Do Formative Assessment 4.5.