Traveling Waves Standing Waves

Musical Instruments Musical Instruments all work by producing standing waves. There are three types of instrument.

Sounds are produced by vibrating matter. strings 2. strings membranes 1. membranes air columns 3. air columns

Standing Waves First some revision on Superposition and Reflection link to applet Look in Applets > Waves > Transverse Waves > Adding Transverse Waves > Reflection and Transmission When you wiggle the end of a spring, a wave travels down the spring, reflects off a fixed end inverted and comes back.

If you send a series of waves with the same frequency and amplitude they can interfere with the reflected waves to set up a standing wave. link to applet Look in Applets > Waves > Transverse Waves > Adding Transverse Waves/ Continuous

This is what happens on a string instrument. e.g

This is what happens at the closed end. Incoming pulse Reflection (if no phase change) Reflection (with phase change) Resultant wave

¼ cycle later. Incoming pulse Reflection (with phase change) Reflection (if no phase change) Resultant wave

Half a cycle later Incoming pulse Reflection (no phase change) Reflection (with phase change) Resultant wave

Resulting motion………… Resultant wave

t = ………

link to standing wave on string link to wave superposition

There are only certain frequencies that a standing wave can have. These are called the harmonics. They are always multiples of the fundamental link to harmonics

1 st harmonic (or fundamental) This is the lowest possible frequency standing wave. It is half a wavelength long. node antinode Nodes are points of zero amplitude Antinodes are points of maximum amplitude

The distance between two adjacent nodes (or antinodes) is always half a wavelength.

1 st harmonic 2nd harmonic 3rd harmonic Wavelength = ……… x L L Frequency = f 1 Wavelength = ……… x L Frequency = ………f 1 L = ……… wavelength

Note that only certain wavelength standing waves are possible, because only a whole number of half wavelengths must fit on the string. The string can only contain: ½ λ or 2(½ λ) or 3(½ λ) …………… ½ λ or 1 λ or 1 ½ λ or 2 λ ……… etc. L

link to standing wave on bridge

Why do different instruments sound different when playing the same note? This is what a guitar string does when it is producing more than one harmonic link: Harmonics on Guitar

Guitar playing “A” Violin playing “A” amplitude frequency amplitude frequency

Why does a guitar playing “A” (440Hz) sound different to a piano playing “A”? Both instruments produce 440 Hz plus lots of higher harmonics.(880, 1320…..) The loudness of each harmonic from the instruments is different. The harmonics add together to give a complex note that is different for each instrument. We hear all the harmonics added.

What Determines the Frequency of the Fundamental? The frequency of a standing wave is given by: The wavelength is: The wavespeed is: The frequency is therefore : How does each variable affect the frequency?

Changing the Frequency 1 Instruments such as harps and pianos have a large number of string of different length and density.

Changing the Frequency 2 Instruments such as guitars usually only have six strings but the musician changes the note by altering the length (and tension)

Longitudinal Standing Waves

This is a sound spectrum of the singing rod. It is 75 cm long. Calculate the speed of sound in aluminium

Wind Instruments A similar thing happens in a tube of air. First of all, the air at one end must be made to vibrate. The vibration causes sound waves to travel down a tube. They reflect off the end and come back. They interfere with the incident waves At certain frequencies, standing waves are set up.

There are two types of tube. OPEN/OPEN and OPEN/CLOSED We will look at OPEN tubes first:

Open/Open Pipe

We can see how the standing waves in an air pipe using a speaker: link to standing wave in pipes

1 st harmonic 2nd harmonic 3rd harmonic Wavelength = ……… x L Frequency = f 1 Wavelength = ……… x L Frequency = ………f 1 L = ……… wavelength L

Note that only certain wavelength standing waves are possible, because only a whole number of half wavelengths must fit in the pipe. The pipe can only contain: ½ λ or 1 λ or 1 ½ λ or 2 λ ……… etc. L

Why do a flute and recorder sound different when playing 440Hz? Both instruments produce the same fundamental plus lots of higher harmonics.(880, 1320…..) Different shapes emphasise different harmonics The loudness of the harmonics from each instrument is different. The harmonics add together to give a complex note that is different for each instrument.

Open / Closed Pipe

1 st harmonic 2nd harmonic 3rd harmonic L

1 st harmonic 3rd harmonic 5th harmonic Wavelength = ……… x L Frequency = f 1 Wavelength = ……… x L Frequency = ………f 1 L = ……… wavelength L

Note that in an open/closed pipe only the odd harmonics are possible. i.e. f 1 3f 1 5f 1 7f 1 This is because the pipe contains

Note that only certain wavelength standing waves are possible, because only a whole number of quarter wavelengths must fit in the pipe. The pipe can only contain: 1/4 λ or 3/4 λ or 5/4 λ or 2 λ ……… etc. L

Comparing the Frequency in a Closed tube and an Open tube 1 st harmonic Wavelength = ……… x L Frequency = f L = ……… wavelength L L Wavelength = ……… x L Frequency = L = ……… wavelength

Harmonics on a Flute and Clarinet FLUTE CLARINET

ex. What is the wavelength of this standing wave? The frequency is 3200 Hz. What is the wave speed? 15 cm

Changing the Frequency. “v” is the speed of the wave in the pipe which is 340 ms -1. So to change the frequency, you change the wavelength.

Changing the Length there are 3 ways to change the length

1. A whole lot of fixed length pipes

2. A single pipe with a variable length.

3. A single pipe with lots of holes to change the “effective length”.

OPEN (Antinode)

extension Rectangular Membrane Applet