 The principle of superposition for two identical waves traveling in opposite directions.  Snapshots of the blue and the green waves and their red sum shows a wave that appears to not be traveling.  Because the resultant red wave is not traveling it is called a standing wave. E E are the result of the interference of two identical waves with the same frequency and the same amplitude traveling in opposite direction. STANDING WAVES

N N NN N N N AAA A A A N N NN AAA A A Nodes and antinodes  Any points where the standing wave has no displacement is called a node (N).  The lobes that grow and shrink and reverse are called antinodes (A).

 You may be wondering how a situation could ever develop in which two identical waves come from opposite directions. Well, wonder no more.  When you pluck a stringed instrument, waves travel to the ends of the string and reflect at each end, and return to interfere under precisely the conditions needed for a standing wave.  Note that there are two nodes and one antinode.  Why must there be a node at each end of the string? L N N A Because it is fixed at each end. Boundary conditions - strings

L Let’s consider a string of length L. Let both ends be fixed. Let us see what shapes can be formed on that string. Boundary conditions - strings

L L L

n = 1, 2, 3… L = n 2 Distance between two nodes is /2 (one pillow is /2) The frequencies at which standing waves are produced are called natural frequencies or resonant frequencies of the string or pipe or... the lowest freq. standing wave is called FUNDAMENTAL or the FIRST HARMONICS The higher freq. standing waves are called HARMONICS (second, third...) or OVERTONES

applet that has everything – standing waves – comparison of transverse and longitudinal waves Standing Waves in a Drum Membrane Standing waves in a drum membrane are complicated and satisfactory analysis requires knowledge of Bessel functions. So, this is just for fun. Mrs. Radja’s fun Standing waves in a chain vertically hung

Mathematicians Pythagoras, Leibniz, Riemann, Fourier are just few names in a long humankind's attempt to understand the mysteries of the music. There is a beautiful simple logic together with aesthetic similarities that are shared by mathematics and music. Why a tuning fork, a violin and a clarinet sound very different, even when they are all playing an A, say? The reason the violin doesn't look and sound like the tuning fork is that it is playing, not just an A, but also a combination of different frequencies called the harmonics.

Overtones – higher harmonics Overtones are the other frequencies besides the fundamental that exist in musical instruments. Instruments of different shapes and actions produce different overtones. The overtones combine to form the characteristic sound of the instrument. For example, both the waves below are the same frequency, and therefore the same note. But their overtones are different, and therefore their sounds are different. Note that the violin's jagged waveform produces a sharper sound, while the smooth waveform of the piano produces a purer sound, closer to a sine wave. Click on each instrument to hear what it sounds like. Keep in mind that all are playing the same note. Overtones are the other frequencies besides the fundamental that exist in musical instruments. Instruments of different shapes and actions produce different overtones. The overtones combine to form the characteristic sound of the instrument. For example, both the waves below are the same frequency, and therefore the same note. But their overtones are different, and therefore their sounds are different. Note that the violin's jagged waveform produces a sharper sound, while the smooth waveform of the piano produces a purer sound, closer to a sine wave. Click on each instrument to hear what it sounds like. Keep in mind that all are playing the same note. to see the way the first five harmonics combine to build up the wave shape created by a violin wave shape created by a violin see the way the first five harmonics combine to build up the wave shape created by a clarinet wave shape created by a clarinet it is important to notice that although these sounds have the same fundamental frequency/pitch each sound sounds different because it is a combination (mixture) of harmonics at different intensities.

Stringed instruments Three types Three types Plucked: guitar, bass, harp, harpsichord Plucked: guitar, bass, harp, harpsichord Bowed: violin, viola, cello, bass Bowed: violin, viola, cello, bass Struck: piano Struck: piano All use strings that are fixed at both ends All use strings that are fixed at both ends Use different diameter strings (mass per unit length is different) Use different diameter strings (mass per unit length is different) The string tension is adjustable - tuning The string tension is adjustable - tuning Vibration frequencies In general, f = v /, where v is the propagation speed of the string In general, f = v /, where v is the propagation speed of the string The propagation speed depends on the diameter (mass per unit length) and tension The propagation speed depends on the diameter (mass per unit length) and tension Modes Modes Fundamental: f 1 = v / 2L Fundamental: f 1 = v / 2L First harmonic: f 2 = v / L = 2 f 1 First harmonic: f 2 = v / L = 2 f 1 The effective length can be changed by the musician “fingering” the strings The effective length can be changed by the musician “fingering” the strings

Sounds may be generally characterized by pitch, loudness (amplitude) and quality. Pitch is perceived freq – determined by fundamental freq. Timbre is that unique combination of fundamental freq and overtones (harmonics) that gives each voice, musical instrument, and sound effect its unique coloring and character. Timbre is that unique combination of fundamental freq and overtones (harmonics) that gives each voice, musical instrument, and sound effect its unique coloring and character. The greater the number of harmonics, the more interesting is the sound that is produced. The greater the number of harmonics, the more interesting is the sound that is produced. Sound "quality“, “color” or "timbre" describes those characteristics of sound which allow the ear to distinguish sounds which have the same pitch and loudness.

A noise is a jumble of sound waves. A tone is a very regular set of waves, all the same size and same distance apart.

Many musical instruments depend on the musician in some way moving air through the instrument. ▪ This includes brass and woodwind instruments, as well as instruments like pipe organs. ▪ All instruments like this can be divided into two categories, open ended or closed ended. ▪ A “pipe” can be any tube, even if it has been bent into different shapes or has holes cut into it.

A closed ended instrument has one end closed off, and the other end open. An example would be an instrument like some organ pipes (although in some designs they are open), or clarinet. Although you blow in through the mouth piece of a flute, the opening you’re blowing into isn’t at the end of the pipe, it’s along the side of the flute. The end of the pipe is closed off near the mouth piece. An open ended instrument has both ends open to the air. An example would be an instrument like a trumpet. You blow in through one end and the sound comes out the other end of the pipe. The keys on the trumpet or flute allow the air to move through the "pipe" in different ways so that different notes can be played.

Remember that it is actually air that is doing the vibrating as a wave here. The air at the closed end of the pipe must be a node (not moving), since the air is not free to move there and must be able to be reflected back. There must also be an antinode where the opening is, since that is where there is maximum movement of the air. The frequencies of sounds made by these two types of instruments are different because of the different ways that air will move at a closed or open end of the pipe.

 We can also set up standing waves in pipes.  In the case of pipes, longitudinal waves are created and these waves are reflected from the ends of the pipe.  Consider a closed pipe of length L which gets its wave energy from a mouthpiece on the left side.  Why must the mouthpiece end be an antinode?  Why must the closed end be a node? Air can’t move. Source: molecules can be displaced by the larges amount here. Boundary conditions – closed pipes

 The IBO requires you to be able to make sketches of string and pipe harmonics (both open and closed) and find wavelengths and frequencies.  In an open-ended pipe you there is an antinode at the open end because the medium can vibrate there (and, of course, antinode at the mouthpiece). Boundary conditions – open pipes

Distinguishing between standing and traveling waves  A standing wave consists of two traveling waves carrying energy in opposite directions, so the net energy flow through the wave is zero. E E

Beats are a periodic variation in loudness (amplitude) – throbbing - due to interference of two tones of slightly different frequency. CLICK Two waves with slightly different frequencies are travelling to the right. The resulting wave travels in the same direction and with the same speed as the two component waves. Producing beats: When two sound waves of different frequency approach your ear, the alternating constructive and destructive interference results in alternating soft and loud sound. The beat frequency is equal to the absolute value of the difference in frequencies of the two waves.

Useful for tuning musical instruments – listen for beats to disappear (when frequency of instrument is identical to a tuning fork) Useful for tuning musical instruments – listen for beats to disappear (when frequency of instrument is identical to a tuning fork) Beats produced when incident wave interferes with a reflected wave from a moving object: reflected wave has Doppler-shifted frequency, so the two waves differ slightly in freq. Hear beats. Beats produced when incident wave interferes with a reflected wave from a moving object: reflected wave has Doppler-shifted frequency, so the two waves differ slightly in freq. Hear beats. That underlies how any instrument that measures speed using ultrasound work – measures beat freq. – gets Doppler shift in frequency which is related to speed of the object. That underlies how any instrument that measures speed using ultrasound work – measures beat freq. – gets Doppler shift in frequency which is related to speed of the object. Also underlies how dolphins (and others) use beats to sense motions Also underlies how dolphins (and others) use beats to sense motions

(1) A violinist tuning her violin, plays her A-string while sounding a tuning fork at concert-A 440 Hz, and hears 4 beats per second. When she tightens the string (so increasing its freq), the beat frequency increases. What should she do to tune the string to concert-A, and what was the original untuned freq of her string? What should she do to tune the string to concert-A, and what was the original untuned freq of her string? Beat freq = 4 Hz, so orig freq is either 444 Hz or 436 Hz. Increasing freq increases beat freq, so makes the difference with concert-A greater. So orig freq must have been 444 Hz, and she should loosen the string to tune it to concert-A. (2) A human cannot hear sound at freqs above Hz. But if you walk into a room in which two sources are emitting sound waves at 100 kHz and 102 kHz you will hear sound. Why? You are hearing the (much lower) beat frequency, 2 kHz = 2000 Hz.

Solving problems involving standing waves PRACTICE: A tube is filled with water and a vibrating tuning fork is held above the open end. As the water runs out of the tap at the bottom sound is loudest when the water level is a distance x from the top. The next loudest sound comes when the water level is at a distance y from the top. Which expression for is correct? A. = x B. = 2x C. = y-x D. = 2(y-x)  v = f and since v and f are constant, so is.  The first possible standing wave is sketched.  The second possible standing wave is sketched.  Notice that y – x is half a wavelength.  Thus the answer is = 2(y - x). y-x

PRACTICE: This drum head, set to vibrating at different resonant frequencies, has black sand on it, which reveals 2D standing waves.  Does the sand reveal nodes, or does it reveal antinodes?  Why does the edge have to be a node? Nodes, because there is no displacement to throw the sand off. The drumhead cannot vibrate at the edge. Solving problems involving standing waves

 Alternate lobes have a 180º phase difference. Solving problems involving standing waves

 Make a sketch. Then use v = f. antinode antinode L / 2 = L v = f = 2L f = v / f = v / (2L) Solving problems involving standing waves

 Reflection provides for two coherent waves traveling in opposite directions.  Superposition is just the adding of the two waves to produce the single stationary wave.

 The figure shows the points between successive nodes.  For every point between the two nodes f is the same.  But the amplitudes are all different.  Therefore the energies are also different. Solving problems involving standing waves

 Energy transfer via a vibrating medium without interruption.  The medium itself does not travel with the wave disturbance.  Speed at which the wave disturbance propagates.  Speed at which the wave front travels.  Speed at which the energy is transferred. Solving problems involving standing waves

 Frequency is number of vibrations per unit time.  Distance between successive crests (or troughs).  Distance traveled by the wave in one oscillation of the source. FYI: IB frowns on you using particular units as in “Frequency is number of vibrations per second.” FYI: There will be lost points, people! Solving problems involving standing waves

 The waves traveling in opposite directions carry energy at same rate both ways. NO energy transfer.  The amplitude is always the same for any point in a standing wave. Solving problems involving standing waves

L P / 4 = L = 4L L Q / 2 = L = 2L v = f f = v / f P = v / (4L) f Q = v / (2L) v = 4Lf P f Q = 4Lf P / (2L) f Q = 2f P Solving problems involving standing waves

 The tuning fork is the driving oscillator (and is at the top).  The top is thus an antinode.  The bottom “wall” of water allows NO oscillation.  The bottom is thus a node. Solving problems involving standing waves

 Sound is a longitudinal wave.  Displacement is small at P, big at Q. Solving problems involving standing waves

 If the lobe at T is going down, so is the lobe at U. Solving problems involving standing waves

 Pattern 1 is 1/2 wavelength.  Pattern 2 is 3/2 wavelength.  Thus f 2 = 3f 1 so that f 1 / f 2 = 1/3. Solving problems involving standing waves

●Particles are “grainy” at the quantum level. ●The “extra” spatial dimensions required by string theory are “curled” in upon themselves and only visible at very small dimensions. ●Superstring theory uses 11- dimensional standing waves as its particle model.