Superposition and Standing Waves
18.1 18.2 18.3 18.4 18.5 18.6 18.7 18.8 Superposition and Interference Standing Waves Standing Waves in a String Fixed at Both Ends Resonance Standing Waves in Air Columns Standing Waves in Rod and Plates Beats: Interference in Time Non-Sinusoidal Wave Patterns
ANSWERS TO QUESTIONS
Q18.1 No. Waves with other waveforms are also trains ofdisturbance that add together when waves from different sources move through the same medium at the same time. The energy has not disappeared, but is still carried by the wave pulses. Each particle of the string still has kinetic energy. This is similar to the motion of a simple pendulum. The pendulum does not stop at its equilibrium position during oscillation—likewise the particles of the stringdo not stop at the equilibrium position of the string when these two waves superimpose. No. A wave is not a solid object, but a chain of disturbance. As described by the principle of superposition, the waves move through each other.
They can, wherever the two waves are nearly enough in phase that their displacements will add to create a total displacement greater thanthe amplitude of either of the two original waves. When two one-dimensional sinusoidal waves of the same amplitude interfere, this condition is satisfied whenever the absolute value of the phase difference between the two waves is less than 120°. When the two tubes together are not an efficient transmitter of sound from source to receiver, they are an efficient reflector. The incoming sound isreflected back to the source. The waves reflected by the two tubes separately at the junction interfere constructively. No. The total energy of the pair of waves remains the same. Energy missing from zones of destructive interference appears in zones of constructive interference. Each of these standing wave patterns is made of two superimposed waves of identical frequencies traveling, and hencetransferring energy, in opposite directions. Since the energy transfer of the waves are equal, then no net transfer of energy occurs. Damping, and non–linear effects in the vibration turn the energy of vibration into internal energy. The air in the shower stall can vibrate in standing wave patterns to intensify those frequencies in your voice which correspond to its free vibrations. The hard walls ofthe bathroom reflect sound very well to make your voice more intense at all frequencies, giving the room a longer reverberation time. The reverberant sound may help you to stay on key. 523
524 Q18.10 Q18.11
Superposition and Standing Waves
The trombone slide and trumpet valves change the length of the air column inside the instrument, to change itsresonant frequencies. The vibration of the air must have zero amplitude at the closed end. For air in a pipe closed at one end, the diagrams show how resonance vibrations have NA distances that are odd integer submultiples of the NA distance in the fundamental vibration. If the pipe is open, resonance vibrations have NA distances that are all integer submultiples of the NA distance in thefundamental.
FIG. Q18.11 Q18.12 What is needed is a tuning fork—or other pure-tone generator—of the desired frequency. Strike the tuning fork and pluck the corresponding string on the piano at the same time. If they are precisely in tune, you will hear a single pitch with no amplitude modulation. If the two pitches are a bit off, you will hear beats. As they vibrate, retune the piano string until thebeat frequency goes to zero. Air blowing fast by a rim of the pipe creates a “shshshsh” sound called edgetone noise, a mixture of all frequencies, as the air turbulently switches between flowing on one side of the edge and the other. The air column inside the pipe finds one or more of its resonance frequencies in the noise. The air column starts vibrating with large amplitude in a standing wave...