Homework problems

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.
2	(a) Using the equations from optional section 4.2, 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 3.0x108 and 2.2x108 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 B-flat clarinet (the most common kind) produces its lowest note, at about 230 Hz, when half of a wavelength fits inside its tube. Compute the length of the clarinet. [Check: The actual length of a clarinet is about 67 cm from the tip of the mouthpiece to the end of the bell. Because the behavior of the clarinet and its coupling to air outside it is a little more complex than that of a simple tube enclosing a cylindrical air column, your answer will be close to this value, but not exactly equal to it.]
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: Eb (150 Hz), Eb (300 Hz), Bb (450 Hz), and Eb (600 Hz). How is this possible? (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 five or six harmonics of C and G, and determine whether these two notes will be consonant or dissonant. (b) Do the same for C and B flat. [Hint: Remember that harmonics that differ by about 1- 10% cause dissonance.]
6	Brass and wind instruments go up in pitch as the musician warms up. Suppose a particular trumpet's frequency goes up by 1.2%. 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 1.88x10 -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. (c) Determine the change in temperature, in units of degrees C, that would account for the observed 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 3.2, 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.