STANDING WAVES are the result of the interference of two identical waves with the same frequency and the same amplitude traveling in opposite direction. E E © 2014 By Timothy K. Lund  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.

Nodes and antinodes Distance between two nodes is l/2 N N N N Any points where the standing wave has no displacement is called a node (N). A A A A A N N N N N N N The lobes that grow and shrink and reverse are called antinodes (A). A A A A A A Distance between two nodes is l/2 © 2014 By Timothy K. Lund

Boundary conditions - strings  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? Because it is fixed at each end. L N N A © 2014 By Timothy K. Lund

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

Boundary conditions - strings L L L

n = 1, 2, 3… L = n l 2 Distance between two nodes is l/2 (one pillow is l/2) 1. harmonic: 𝐿= λ 2 → λ=2𝐿 → 𝑓 1 = v 𝜆1 = v 2L 2. harmonic: 𝐿=λ → λ=𝐿 → 𝑓 2 = v 𝜆2 = v L =2 v 2L 3. harmonic: 𝐿=3 λ 2 → λ= 2 3 𝐿 → 𝑓 3 = v 𝜆3 = 3v 2L =3 v 2L 4. harmonic: 𝐿=2λ → λ= 𝐿 2 → 𝑓 4 = v 𝜆4 = 2v L =4 v 2L 𝑓 𝑛 =𝑛 v 2L =𝑛 𝑓 1 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

A noise is a jumble of sound waves 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.

Boundary conditions – closed pipes  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? 1. harmonic: 𝐿= λ 1 4 → λ 1 =4𝐿 → 𝑓 1 = v 𝜆1 = v 4L © 2014 By Timothy K. Lund 2. harmonic: 𝐿=3 λ 2 4 → λ 2 = 4𝐿 3 → 𝑓 2 = v 𝜆2 = 3v 4L 3. harmonic: 𝐿=5 λ 3 4 → λ 3 = 4𝐿 5 → 𝑓 3 = v 𝜆3 = 5v 4L Source. Air can’t move.

Boundary conditions – open pipes 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). 1. harmonic: 𝐿= λ 1 2 → λ 1 =2𝐿 → 𝑓 1 = v 𝜆1 = v 2L 2. harmonic: 𝐿= λ 2 → λ 2 =𝐿 → 𝑓 2 = v 𝜆2 = 2v 2L © 2014 By Timothy K. Lund 3. harmonic: 𝐿=3 λ 3 2 → λ 3 = 2𝐿 3 → 𝑓 3 = v 𝜆3 = 3v 2L The IBO requires you to be able to make sketches of string and pipe harmonics (both open and closed) and find wavelengths and frequencies.

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. © 2014 By Timothy K. Lund

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? y-x 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).

Solving problems involving standing waves 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. © 2014 By Timothy K. Lund The drumhead cannot vibrate at the edge.

Solving problems involving standing waves © 2014 By Timothy K. Lund Alternate lobes have a 180º phase difference.

Solving problems involving standing waves Make a sketch. Then use v = f. antinode antinode L © 2014 By Timothy K. Lund  / 2 = L v = f  = 2L f = v /  f = v / (2L)

Solving problems involving standing waves © 2014 By Timothy K. Lund 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.

Solving problems involving standing waves © 2014 By Timothy K. Lund 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. © 2014 By Timothy K. Lund 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. FYI: IB frowns on you using particular units as in “Frequency is number of vibrations per second.” FYI: There will be lost points, people! © 2014 By Timothy K. Lund Distance between successive crests (or troughs). Distance traveled by the wave in one oscillation of the source.

Solving problems involving standing waves © 2014 By Timothy K. Lund 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 P / 4 = L Q / 2 = L © 2014 By Timothy K. Lund  = 4L  = 2L fQ = v / (2L) v = f fP = v / (4L) fQ = 4LfP / (2L) f = v /  v = 4LfP fQ = 2fP

Solving problems involving standing waves The tuning fork is the driving oscillator (and is at the top). The top is thus an antinode. © 2014 By Timothy K. Lund The bottom “wall” of water allows NO oscillation. The bottom is thus a node.

Solving problems involving standing waves © 2014 By Timothy K. Lund 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. © 2014 By Timothy K. Lund

Solving problems involving standing waves © 2014 By Timothy K. Lund Pattern 1 is 1/2 wavelength. Pattern 2 is 3/2 wavelength. Thus f2 = 3f1 so that f1 / f2 = 1/3.

Beats are a periodic variation in loudness (amplitude) – throbbing - due to interference of two tones of slightly different frequency. 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. CLICK The beat frequency is equal to the absolute value of the difference in frequencies of the two waves. 𝑓= 𝑓 1 − 𝑓 2

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. 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

(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? 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 20 000 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.

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 applet that has everything – standing waves – comparison of transverse and longitudinal waves Standing waves in a chain vertically hung

Why a tuning fork, a violin and a clarinet sound very different, even when they are all playing an A, say? 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. 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.  see the way the first five harmonics combine to build up the wave shape created by a clarinet to see the way the first five harmonics combine to build up the wave shape created by a violin 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.

Vibration frequencies Stringed instruments Three types Plucked: guitar, bass, harp, harpsichord Bowed: violin, viola, cello, bass Struck: piano All use strings that are fixed at both ends Use different diameter strings (mass per unit length is different) The string tension is adjustable - tuning Vibration frequencies 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 Modes Fundamental: f1 = v / 2L First harmonic: f2 = v / L = 2 f1 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.  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.