Wave Interference Practice Problems

Two speakers are placed facing an audience. The speakers are playing a tone in phase with each other at a wavelength of 0.85 m. A member of the audience has the position shown. Determine the path difference for the waves emitted by the speakers. 

Mobile phone network towers P and Q are separated by 375 m. Tower P is located left of tower Q. Their antennas are radiating in phase at identical frequencies. A mobile phone is located to the left of antenna P at point R, 1.2 km from the tower on the axis connecting the two antennas. Mobile networks have various bands with different frequencies and wavelengths. Assuming the bands can have any frequency value, determine the largest wavelength that produces constructive interference at point R. 

Digital television broadcast towers F and G are 257 m apart. Tower F is located left of tower G. Their antennas radiate in phase at the same frequency. A TV antenna is located at point H, 1 km to the right of antenna G on the axis connecting the two antennas. TV networks broadcast at various frequencies called channels. Assuming a channel can have any frequency, and similarly, the wavelength, determine the longest wavelength of a channel that will cause destructive interference at point H. 

Two speakers in an open field are separated by 20.0 m. A tune of 550 Hz from the same source is channeled to the speakers. The temperature is quite high, raising the speed of sound in air to 349 m/s. A recording microphone is located at the center of the two speakers. What distance measured from the midpoint of the two speakers will the microphone detect a first maximum gain of the sound? 

Two speakers on a roadshow truck are 11.0 m apart. A notification of frequency 850 Hz and in the same phase is playing from the speakers. The speed of sound in the air at that instant is 346 m/s. The reflections from the walls of the truck are negligible. A programmer is standing at the center of the two speakers listening carefully while directly facing one of the speakers. This orientation means both ears have the same distance measured from both speakers. Does the programmer detect constructive or destructive interference? Explain?

A 75 inch  LED TV has left and right stereo speakers separated by 1.50 m. The speakers, which are run by the same sound card inside the TV are beeping the same notification tone. The microphone on a recording smartphone is placed perpendicularly and 0.8 m directly in front of one of the speakers. Determine all frequencies between 20 Hz and 1.5 kHz that cause constructive interference at the location of the microphone.

A vehicle (at rest) is fitted with two sirens that are 2.2 m apart. The sirens are operated by the same circuitry, emitting pure sine waves that are in phase. A point S lies 1.1 m on one side measured from the sirens. For sound waves traveling straight from the siren to point S, calculate the least frequency that causes destructive interference at point S.

Two identical alarm speakers connected to the same circuit are fitted to a building. They are separated by 5.0 m. Point P is located 2.0 m from alarm D on the axis connecting the two speakers. If pure sine waves emitted by both alarms are in phase and move straight to point P, determine the least frequency capable of causing constructive interference at point P.

A radio with two detached speakers is tuned to a radio station playing an alert tone at 1050 Hz. The speakers are initially placed at the same point, 3.20 m away from a listener. One speaker is moved backward from the listener. What distance x between the two speakers will create destructive interference for the second time at the listener's position?

Consider a circular wave with a frequency of 100 Hz propagating in a medium with a speed of 340 m/s. At a certain instant, the phase at a distance of 2.0 m from the source is 3π rad. What is the phase at distances of i) 1.5 m and ii) 2.5 m from the source at the same instant?

A radio station transmits a signal at a frequency of 98.5 kHz. If the phase difference between the two antennas which are located along the x-axis is 2π/3 rad, what is the distance between the antennas? Assume that the speed of the signal is 3.00 × 10⁶ m/s.

Two speakers are placed along the x-axis, one at x= -100 m and the other at x = +100 m. The speakers emit sound waves with a frequency of 127.5 Hz. A person stands at the point (x,y) = (100 m, 600 m). Determine whether there will be perfectly constructive interference, perfectly destructive interference or something in between at that point. The phase difference between the two speakers is 2π/3 rad. Take the speed of sound in air as 340 m/s. For simplicity, consider only 1 decimal place.

Two audio emitters are separated by 1.20 m. They emit sound at a wavelength of 52.0 cm. A sound recorder orbits one of the emitters in a circular path at a radius of 1.35 m. Determine all values of d less than or equal to 1.35 m measured from the other emitter for which the recorder detects maximum gain in the audio recorded. 

Two radio speakers are separated by 2.00 m. The speakers are playing a note of wavelength 68.6 cm that spreads evenly in all directions. A drone carrying a microphone orbits one of the speakers at a radius of 2.20 m. Determine all values of d less than or equal to 2.20 m measured from the other speaker for which the microphone will not detect any sound.

During a concert, three violin artists are playing the same tune on stage. The placement of the violin artists is depicted in the figure. You are sitting in the audience 8.0 m away in front of the middle artist. Determine the minimum distance through which the middle violin artist must change his position backward so that you can hear maximum sound intensity. Consider the amplitude of the sound is 'A' and the frequency is 425 Hz from each violin. The velocity of sound is 340 m/s.

In an annual gathering of a college, three students are doing a piano show. They are sitting as portrayed in the figure. The college principal is sitting  3.0 m away in front of the middle student playing the piano. The amplitude of the sound played is 'B' at a frequency of 170 Hz by each student. Determine the ratio of the maximum intensity of sound to the sound intensity of a single piano if the sound waves emitted are in phase. Consider the speed of sound is 340 m/s and all the sound measurements are done at the position of Principal.

Two stones are dropped and create a ripple into a calm pond as illustrated below. Determine the type of interference that would be observed at points M, N, and O.

Consider two mechanical waves traveling along a string, generated by two sources that vibrate in phase, creating waves with the same amplitude and frequency. One of the waves has a phase shift relative to the other. They can be represented by:

$\begin{array}{l}W_1=A\sin (kx-ωt)\\ W_2=A\sin (kx-ωt+φ)\end{array}$

The resultant wave can be expressed as:

$W=2A\cos (φ/2)\sin (kx-ωt+φ/2)$

Depict the resultant wave, using both an equation and a verbal explanation, when $φ=\pi /2$.

Two violinists are practicing the same note at a frequency of 350 Hz and with the same amplitude. However, one of them is slightly out of sync and plays the note 180° out of phase. If they stand side by side, what is the minimum distance they should be apart for complete destructive interference to occur? Assume T = 22°.

The two water pulses shown in the figure below are moving toward each other. Sketch the shape of the water surface at the moment they directly overlap.

Two transverse pulses are traveling towards each other through a slinky. Pulse A is moving to the right, and Pulse B is moving to the left. When the pulses intersect, the slinky is completely straight. Discuss what happens to the energy of the pulses at this point.

Consider two water waves generated by two sources that vibrate in phase, creating waves with the same amplitude and frequency. One of the waves has a phase shift relative to the other. They can be represented by:

W₁ = D sin(kx - ωt)

W₂ = D sin(kx - ωt + φ)

The resultant wave can be expressed as W = (2D cos(φ/2)) sin(kx - ωt + φ/2). Determine the amplitude of the resulting water wave. Is the wave purely sinusoidal?

When studying water wave interference, one can observe that destructive interference takes place if the waves are out of phase by 1/2 wavelength or 180°. Explain why a 180° phase difference corresponds to 1/2 wavelength in the context of water waves.

On a string, as shown, two wave pulses move with equal speed of 8.0 cm/s towards each other. At the start of the observation, the pulses are 20 cm apart between their peaks. Draw the wave pulses' positions at t = 1.0 s, t = 1.5 s, and t = 2.0 s.

A teacher uses two synchronous sound emitters placed on a graduated ruler and a sound detector to show her students the interference phenomenon. The sound detector is placed in a fixed position to the right of the two emitters on the graduated ruler. When the distance between the two emitters is 15 cm, the sound detector measures the maximum intensity. The teacher enhances the distance separating the two emitters until zero intensity is measured. The distance separating the two emitters is now 45 cm. Calculate the wavelength of the sound wave produced by the emitters.

Two sound emitters having π rad phase difference are placed in a 25°C hall on a horizontal graduated ruler. The frequency of the two emitters is 485 Hz. A microphone probe is placed in a fixed position to the right of the two emitters on the graduated ruler. Calculate the minimum distance that should separate the two sound emitters in order to measure, with the microphone probe, a constructive interference.

The refractive index of a newly developed material can be approximated by the equation n(λ) = A/λ^0.5, where λ is the wavelength of light in nanometers and A is a constant equal to 37.6 nm^0.5. A 200 nm thick layer of the material is deposited on a glass substrate (n = 1.55). Determine the wavelengths of visible light that exhibit constructive interference between the reflected waves from the top and bottom interfaces of the thin layer.

Two identical tuning forks P and Q, placed at the same position, are playing sounds of frequency 650 Hz. A person is 6.50 m away to the right from this initial position. The tuning fork P is gradually moved away to the left while Q is kept still, as shown in the figure. For a certain position of the tuning fork P, the person first hears a destructive interference. Determine the distance between P and Q for which the next destructive interference will be heard. Assume that the speed of sound in air is 343 m/s.

Two synchronized radio towers, Tower 1 and Tower 2, are transmitting signals at the same frequency of 400 Hz. Tower 2 is located exactly 1.0 m north of Tower 1. An observer is located somewhere along the line joining these two towers. If 'y' represents the distance from the observer to Tower 1, then find all possible values for y which would result in destructive interference at the observer's location. Consider the speed of sound in air to be 340 m/s.

A taut rope, as shown below, has two pulses moving towards each other at a speed of 2.0 cm/s. Sketch the shape of the rope at the end of 3.0 seconds.