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## Week 2

• Homework: Waves and resonance problems

### Laboratory 2: Resonance, Impedance, Time/Frequency Analysis

• View standing-wave motion on a cello string using Max Mathews' driven string system. Measure (roughly) the speed of wave propagation on the string based on the frequency (read from the oscillator) and wavelengh of one or more observed resonances.

• A sound system is setup in the CCRMA recording studio which, through wave interference, causes pressure nodes and antinodes to occur at various locations within the room. Move around the room to find locations where these sound pressure nodes and antinodes occur. Change the sounding frequency to see how this affects the locations of the nodes and antinodes.

• (From Benade, Fundamentals of Musical Acoustics, pg. 171) When one strikes the G4 key of a well-tuned piano, the strings vibrate with the following list of harmonically related characteristic frequencies: 392, 784, 1176, 1568, 1960, 2352, ...Hz (ignoring the slight inharmonicity of real strings). All of these components serve to drive the soundboard, and thence everything else that it is connected to, including the strings. If we slowly but fully depress the key for G3 in order to raise the damper for that note without striking the strings, we supply ourselves with a system whose natural frequencies are as follows: 196, 392, 588, 784, 980, 1176, ...Hz. Striking the G4 key (while the G3 key is still being silently depressed) and releasing it quickly will provide a short burst of excitation to the even-numbered modes of the G3 strings, and these will continue to sing after the direct sound of the G4 strings has been killed off by their damper touching them on the release of the key. Why do we hear a pitch belonging to G4 coming from the open G3 string in this experiment? Reversing the experiment, so that the G4 key is held down and the G3 key is struck briefly, produces a closely similar result, with a similar explanation. Your understanding of what is going on here will be improved if you observe what happens when G3 is replaced by F3 or G3, with the G4 key being used as before.

• Measure the ``impulse response'' of a cylindrical pipe and view the associated ``impedance'' using snd. The impulse response can be roughly measured by quickly slapping a hand over one end of the pipe while recording the response from the other end. The impedance is given by the Fourier transform of the impulse response. Increase the FFT size appropriately to obtain a single transform of the entire impulse response. Repeat the technique, but remove your hand after the slap to obtain a measure of the open-open response of the pipe. What does the impedance tell you for each case? Describe the difference between the two measured impedances. Calculate the theoretical resonances for each case (measure the length of the pipe) and compare them to the recorded values.

• Record a few hard striking sounds (impulsive sounds) and a few ``steady-state'' sounds (whistling or singing a steady tone), being careful not to ``clip'' the signals during the recording process. View the time-domain waveforms and their associated time-varying spectra using snd. What general observation can be made regarding the frequency content of these two types of sounds? Also, consider aspects of partial ratios, partial strengths, and partial decay rates.

• View traveling- and standing-wave motion on a string using the Matlab script traveling.m