Two in-phase speakers produce sound of wavelength \(0.50\,\mathrm{m}\). At a point, the path difference is \(1.0\,\mathrm{m}\). Is the interference constructive or destructive?

Two in-phase speakers produce sound of wavelength \(0.60\,\mathrm{m}\). At a point, the path difference is \(0.30\,\mathrm{m}\). Is the interference constructive or destructive?

A sound has frequency \(680\,\mathrm{Hz}\) in air where \(v=340\,\mathrm{m\,s^{-1}}\). Find its wavelength.

A path difference is \(0.25\,\mathrm{m}\) and the sound wavelength is \(0.50\,\mathrm{m}\). Find the phase difference in radians.

Two coherent in-phase speakers emit \(500\,\mathrm{Hz}\) sound. At a point, the path lengths are \(3.40\,\mathrm{m}\) and \(4.08\,\mathrm{m}\). Use \(v=340\,\mathrm{m\,s^{-1}}\) to classify the interference.

For equal sound amplitudes \(A\) and phase difference \(\Delta\phi=\pi/3\), find the resultant amplitude in terms of \(A\).

For equal sound amplitudes \(A\), what phase difference gives complete cancellation?

Two in-phase speakers are separated by \(2.0\,\mathrm{m}\). A listener is \(5.0\,\mathrm{m}\) from one speaker and \(5.5\,\mathrm{m}\) from the other. For wavelength \(1.0\,\mathrm{m}\), classify the interference.

Two in-phase speakers emit \(1000\,\mathrm{Hz}\) sound in air. Find the smallest nonzero path difference that gives destructive interference.

Two coherent equal-amplitude waves have path difference \(\lambda/4\). Find the resultant amplitude in terms of one-wave amplitude \(A\).

Two in-phase speakers emit \(850\,\mathrm{Hz}\) sound. A listener moves to a point where the path difference is \(0.60\,\mathrm{m}\). Use \(v=340\,\mathrm{m\,s^{-1}}\) to classify the interference and give the order if constructive.

At a point, two equal-amplitude coherent sound waves have phase difference \(120^\circ\). Find the resultant amplitude and intensity ratio compared with one wave alone.

A noise-canceling speaker emits sound of equal amplitude and half-cycle phase shift relative to incoming noise at the ear. Explain the interference condition and the ideal result.

Two speakers emit different frequencies, \(500\,\mathrm{Hz}\) and \(700\,\mathrm{Hz}\). Explain why they do not make a fixed constructive/destructive spatial interference pattern.

For two in-phase coherent speakers, derive the constructive and destructive path-difference conditions from \(\Delta\phi=2\pi\Delta r/\lambda\).

Two equal-amplitude coherent waves meet with path difference \(\Delta r\). Derive the resultant amplitude in terms of \(A\), \(\Delta r\), and \(\lambda\).

A listener receives sound from two in-phase speakers. One path is fixed at length \(r_1\), and the other path length \(r_2\) is adjustable. Derive all values of \(r_2-r_1\) that produce destructive interference, and state the physical restriction on the result.

Two coherent speakers emit equal amplitudes, but speaker B has an initial phase lead \(\phi_0\). Derive the condition on path difference for constructive interference at a listener.

Two equal-amplitude coherent sound waves meet at a point. Derive the phase differences for which the intensity is half of the maximum possible intensity at that point.

A listener reports a minimum that is not silent when two coherent speakers are adjusted for destructive interference. Give a model-based explanation using unequal amplitudes, and derive the minimum resultant amplitude for amplitudes \(A_1\) and \(A_2\).