20.5 Summary by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.
reflection — the bouncing back of part of a wave from a boundary
transmission — the continuation of part of a wave through a boundary
absorption — the gradual conversion of wave energy into heating of the medium
standing wave — a wave pattern that stays in one place
λ — wavelength (Greek letter lambda)
Whenever a wave encounters the boundary between two media in which its speeds are different, part of the wave is reflected and part is transmitted. The reflection is always reversed front-to-back, but may also be inverted in amplitude. Whether the reflection is inverted depends on how the wave speeds in the two media compare, e.g., a wave on a string is uninverted when it is reflected back into a segment of string where its speed is lower. The greater the difference in wave speed between the two media, the greater the fraction of the wave energy that is reflected. Surprisingly, a wave in a dense material like wood will be strongly reflected back into the wood at a wood-air boundary.
A one-dimensional wave confined by highly reflective boundaries on two sides will display motion which is periodic. For example, if both reflections are inverting, the wave will have a period equal to twice the time required to traverse the region, or to that time divided by an integer. An important special case is a sinusoidal wave; in this case, the wave forms a stationary pattern composed of a superposition of sine waves moving in opposite direction.
Key
√ A computerized answer check is available online.
∫ A problem that requires calculus.
⋆ A difficult problem
1. Light travels faster in warmer air. Use this fact to explain the formation of a mirage appearing like the shiny surface of a pool of water when there is a layer of hot air above a road. (For simplicity, pretend that there is actually a sharp boundary between the hot layer and the cooler layer above it.)
2. (a) Using the equations from optional section 20.2, compute the amplitude of light that is reflected back into air at an air-water interface, relative to the amplitude of the incident wave. The speeds of light in air and water are 3.0×108 and 2.2×108 m/s, respectively.
(b) Find the energy of the reflected wave as a fraction of the incident energy. [Hint: The answers to the two parts are not the same.] √
3. A concert flute produces its lowest note, at about 262 Hz, when half of a wavelength fits inside its tube. Compute the length of the flute.
4. (a) A good tenor saxophone player can play all of the following notes without changing her fingering, simply by altering the tightness of her lips: E♭ , E♭ (300 Hz), B♭ (450 Hz), and E♭ (600 Hz). How is this possible? (I'm not asking you to analyze the coupling between the lips, the reed, the mouthpiece, and the air column, which is very complicated.)
(b) Some saxophone players are known for their ability to use this technique to play “freak notes,” i.e., notes above the normal range of the instrument. Why isn't it possible to play notes below the normal range using this technique?
5. The table gives the frequencies of the notes that make up the key of F major, starting from middle C and going up through all seven notes.
(a) Calculate the first four or five harmonics of C and G, and determine whether these two notes will be consonant or dissonant. (Recall that harmonics that differ by about 1-10% cause dissonance.)
(b) Do the same for C and B♭.
6. Brass and wind instruments go up in pitch as the musician warms up. As a typical empirical example, a trumpet's frequency might go up by about 1%. Let's consider possible physical reasons for the change in pitch. (a) Solids generally expand with increasing temperature, because the stronger random motion of the atoms tends to bump them apart. Brass expands by 1.88×10^(-5) per degree C. Would this tend to raise the pitch, or lower it? Estimate the size of the effect in comparison with the observed change in frequency. (b) The speed of sound in a gas is proportional to the square root of the absolute temperature, where zero absolute temperature is -273 degrees C. As in part a, analyze the size and direction of the effect.
7. Your exhaled breath contains about 4.5% carbon dioxide, and is therefore more dense than fresh air by about 2.3%. By analogy with the treatment of waves on a string in section 19.2, we expect that the speed of sound will be inversely proportional to the square root of the density of the gas. Calculate the effect on the frequency produced by a wind instrument.
20.5 Summary by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.