Set 5

(d) all of the above

For standing waves to be produced, the traveling wves (component waves) have to have the same speed, wavelength, amplitude, and frequency.  The only difference is that they travel in opposite directions.

The order of the harmonic represents the number of loops in the standing wave.  Each loop represents 1/2 of a wavelength.  Therefore the third harmonic has three loops equalling a total of (3/2) wavelengths.

Remember, that the order of the harmonic for a fixed-end reflection represents the number of loops.  For fixed-end reflections, the number of loops times the fundamental frequency equals the frequency of the desired harmonic.

For the first harmonic, there is only one loop in the resonance waveform.  Since one loop equals 1/2 wavelength, 1/2 l = 3 meters ® l = 6 meters.

For the third harmonic, there are three loops in the resonance waveform.  Since one loop equals 1/2 wavelength, 3(1/2) l = 3 meters ® l = 2 meters.

All permitted resonance states for fixed-end reflections have a whole number of loops in their waveforms.  To answer this question we need to determine the length of a loop for each requested frequency.  Use the equation v_w = fl to find each wavelength.

v_w = fl ® l = 12/15 = 0.8 meters and  v_w = fl ® l = 12/20 = 0.6 meters

In the first case for 15 hz, a loop would be 0.4 meters.  If 0.4 meters divides evenly into the length of the string then it will be a permitted resonance state.  4 meters / 0.4 meters per loop ® 10 loops.

In the first case for 20 hz, a loop would be 0.3 meters.  If 0.3 meters divides evenly into the length of the string then it will be a permitted resonance state.  4 meters / 0.3 meters per loop = 13.3 loops which is not a wholoe number.  Therefore 20 hz is not a permitted frequency.

Each wave has a wavelength of 0.80 meters and a speed of 250 m/sec. That means that the fundamental frequency of each wave can be found with the equation v_w = fl ® f = 250/0.8 ® f = 31.25 hz 

Since the string is 2 meters long, and 0.8 meters divides into 2 meters 2.5 times, that means that there are 5 loops in the waveform. The order is therefore 5 and the frequency of this resonance state is 5 * 31.25 = 156.25 hz

1.5 segments means 1.5 loops.  Since each loop is half of a wavelength, the length of the antenna holds (1.5)(1/2l) = 0.75l.    Setting 3 meters = 0.75l allows us to determine that the wavelength for this resonance state is 4.0 meters. 

The waveform for the fundamental frequency will be half of loop since one end of the antenna is fixed to the car and the other end is "free."  Setting 0.25l = 3 meters yields a fundamental wavelength of 12 meters.