Two-Speaker Interference Near the Speakers
A student sets up two loudspeakers connected to the same signal generator. The generator frequency is 680 Hz, and the speed of sound in the room is 340 m/s. The speakers are 1.50 m apart and face the same direction. The student walks along a straight line 0.60 m in front of the speakers, parallel to the line joining them, from a point directly in front of one speaker to a point directly in front of the other. The student hears a series of loud and quiet patches. The room has hard walls, so reflected sound may reach the student. The student counts the number of loud patches and finds it is not exactly the number predicted by a simple two-source interference model. The student claims that doubling the frequency to 1360 Hz will double the distance between adjacent loud patches, and that the hard walls have no effect because sound waves do not reflect from walls at these frequencies.
1. Calculate the wavelength of the sound and predict how many loud patches the student should hear along the line between the points directly in front of each speaker.
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Wavelength λ = v/f = 340/680 = 0.50 m. Loud patches occur where the path difference from the two speakers is a whole number of wavelengths: Δd = nλ. At the midpoint of the line, the distances are equal, so Δd = 0 (n = 0). At a point directly in front of one speaker, the distances are 0.60 m and √(0.60² + 1.50²) = √(0.36 + 2.25) = 1.616 m, so the path difference is 1.016 m. This is about 2.03λ, so n = 0, ±1, ±2 can occur within the line. That gives five loud patches if the endpoints are counted: one central maximum and two on each side. A simple model therefore predicts five maxima, but the exact count depends on whether the endpoints fall exactly on a maximum and on reflections.
2. Evaluate the student's claim that doubling the frequency doubles the spacing between adjacent loud patches.
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Adjacent maxima satisfy Δd = nλ and (n + 1)λ, so their spacing along a line depends on λ. If frequency doubles, λ = v/f halves. The path-difference pattern in space is unchanged, so the positions where Δd equals a whole number of wavelengths become closer together: the spacing between adjacent maxima is approximately halved, not doubled. The student has the factor inverted. The exact spacing is not simply proportional to λ everywhere because the path difference changes non-linearly along the line, but the trend is clear: higher frequency gives more closely spaced maxima.
3. Explain why the student counts more loud patches than predicted, and whether the hard walls make the simple two-source model unreliable.
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The simple model assumes only the direct sound from the two speakers. Hard walls reflect sound, so the student receives additional waves with extra path lengths. These reflected waves can interfere constructively or destructively with the direct waves, producing extra maxima and minima and shifting the positions of the expected ones. The room is therefore not a clean two-source interference field. The discrepancy does not mean the wave model is wrong; it means the model is incomplete. To test the two-source prediction better, the student could use absorptive material on the walls, take measurements close to the speakers, or use a microphone and plot intensity rather than relying on hearing.