University physics waves and electricity
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University Physics: Waves and Electricity. Ch1 7 . Longitudinal Waves. Lecture 4. Dr.-Ing. Erwin Sitompul. 2013. Homework 3 : Standing Waves.

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University Physics: Waves and Electricity

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University Physics: Waves and Electricity

Ch17. Longitudinal Waves

Lecture 4

Dr.-Ing. Erwin Sitompul


Homework 3: Standing Waves

Two identical waves (except for direction of travel) oscillate through a spring and yield a superposition according to the equation

(a)What are the amplitude and speed of the two waves?

(b)What is the distance between nodes?

(c)What is the transverse speed of a particle of the string at the position x = 1.5 cm when t = 9/8 s?

Solution of Homework 3

(a)Identical except direction of travel ► standing waves:

(b)Distance between nodes:

Solution of Homework 3

(c)Transversal speed:

At x = 1.5 cm = 15 mm and t = 9/8 s = 3/160 min,

Sound Waves

  • From previous chapter we know that mechanical waves are classified into transverse waves and longitudinal waves.

  • In this class, a sound wave is defined roughly as a longitudinal waves.

  • The figure above illustrates several ideas useful for the next discussions.

  • Point S represents a tiny sound source, called a point source. It emits sound waves in all directions.

  • Wavefronts are surfaces over which the oscillation due to the sound wave have the same value.

  • Rays are directed lines perpendicular to the wavefronts that indicate the direction of travel of the wavefronts.

Sound Waves

  • As a longitudinal wave, sound wave travels through a medium (solid, liquid, or gas), involving oscillations parallel to the direction of wave travel.

  • When a sound wave moves in time, the displacement of air molecules, the pressure, and the density vary sinusoidally with the frequency of the vibrating source.

The Speed of Sound

  • The speed of sound, as also all other mechanical waves, transverse or longitudinal, depends on both an inertial property of the medium (to store kinetic energy) and an elastic property of the medium (to store potential energy).

  • As a sound wave passes through air, potential energy is associated with periodic compressions and expansions of small volume elements of the air.

The Speed of Sound

  • The speed of sound through several mediums are shown as follows.

Traveling Sound Waves

  • Here we examine the displacements and pressurevariations associated with a sinusoidal sound wave traveling through air.

  • The figure below displays such a wave traveling rightward through a long air-filled tube.

  • For a thin element of air of thickness Δx, as the wave travels through this portion of the tube, the element of air oscillates left and right in a simple harmonic motion about its equilibrium position.

Traveling Sound Waves

  • We choose to use a cosine function to show the displacements:

  • As the wave moves, the air pressure at any position x also varies sinusoidally. To describe this variation, we write

Traveling Sound Waves

  • The next figure shows plots of displacement amplitude and pressure amplitude, at t = 0.

  • With time, the two curves would move rightward along the horizontal axes.

  • How?

  • Note that the displacement and pressure variation are π/2 rad (or 90°) out of phase.

  • Thus, for example, the pressure variation Δp at any point along the wave is zero when the displacement here is at maximum.

  • In other time, the pressure at a certain point is at maximum when the displacement there is zero.


  • Like transverse waves, sound waves can undergo interference.

  • Now we will consider, in particular, the interference between two identical sound waves traveling in the same direction.

  • Two point sources S1 and S2 emit sound waves that are in phase and of identical wavelength λ.

  • Thus, as the waves emerge from the sources, their displacements are always identical.

  • The waves travels through point P, with the distance L1 or L2 much greater than the distance between the sources, S1 and S2.

  • The two sources can be approximated to travel in the same direction at P.


  • From the figure, the path L2 traveled by the wave from S2 is longer than the path L1 traveled by the wave from S1.

  • The difference in path lengths means that the waves may not be in phase at point P.

  • The phase difference Φat P depends on their path length difference,

  • The relation between phase difference to path length difference, as we recall from previous chapter, is:


  • Fully constructive interference occurs when Φis zero, 2π, or any integer multiple of 2π.

  • Fully constructive interference

  • Fully destructive interference occur when Φis an odd multiple of π,

  • Fully destructive interference


Fully constructive,arrive “in phase”

Fully destructive,arrive “out of phase”

Example: Interference

Two point sources S1 and S2, which are in phase and separated by distance D = 1.5λ, emit identical sound waves of wavelength λ.

(a)What is the path length difference of the waves from S1 and S2 at point P1, which lies on the perpendicular bisector of distance D, at a distance greater that D from the sources? What type of interference occurs at P1?

The waves undergofully constructive interference at P1

Example: Interference

(b)What are the path length difference and type of interference at point P2?

The waves undergo fully destructive interference at P2

Example: Interference

(c) The figure below shows a circle with a radius much greater than D, centered on the midpoint between sources S1 and S2. What is the number of points N around this circle at which the intereference is fully constructive?

Using the symmetry, as we go around the circle, we will find 6points where the interference is a fully constructive interference

Intensity and Sound Level

  • There is more to sound than frequency, wavelength, and speed. We are well with something called intensity, whether the sound is loud or soft.

  • The intensity I of a sound wave at a surface is the average rate per unit area at which enery is transferred by the wave through or onto the surface.

  • where P is the time rate of energy transfer (the power) of the sound wave and A is the area of the surface intercepting the sound.

  • The intensity I of a sound wave at a surface is the average rate per unit area at which enery is transferred by the wave through or onto the surface.

Variation of Intensity with Distance

  • How intensity varies with distance depends mostly on the shape and orientation of the sound source. Environment usually produces echoes (reflected sound waves) that overlap the direct sound wave.

  • For simplification, we assume that the sound source is a point source that emits the sound isotropically – that is, with equal intensity in all directions.

  • The intensity Iat the sphere with a distance r from the sound source must then be

  • A point source S emits sound waves uniformly in all direction

  • The waves pass through an imaginary sphere of radius r that is centered on S.

The Decibel Scale

  • Human ear can bear the displacement amplitude that ranges from about 10–5 m for the loudest torelable sound to about 10–11 m for the faintest detectable sound.

  • The ratio between the highest and the lowest amplitude is 106.

  • To deal with such an enormous range of values, people use logarithmic scale instead of linear scale.

  • β is called the sound level.

  • dB is the abbreviation for decibel, the unit of sound level.

  • I0 is a standard reference intensity 10–12 W/m2, chosen because it is near the lower limit of human range of hearing.

Intensity and Sound Level

  • β increases by 3 dB every time the sound intensity and sound power is doubled (increases by a factor of 2).

  • β increases by 10 dB every time the sound intensity and sound power increases by an order of magnitude (increases by a factor of 10).

  • In order to be clearly understood, one must speak with sound intensity 10 dB greater than the sound intensity of the background noise.

  • Quiet classroom  40 dB adequate

  • Noisy classroom  70 dB, or with the power 1000x greater

Intensity and Sound Level

  • The minimum audibility level is a function of audible frequency

  • Low and high frequencies require high sound level in order to be audible.

Example: Electric Spark

An electric spark jumps along a straight line of length L = 10 m, emitting a pulse of sound that travels radially outward from the spark. (The spark is said to be a line source of sound.) The power of the emission is Ps = 1.6×104 W.

(a)What is the intensity I of the sound when it reaches a distance r = 12 m from the spark?

(b)At what time rate Pd is sound energy intercepted by an acoustic detector of area Ad = 2.0 cm2, aimed at the spark and located a distance r = 12 m from the spark?

Example: Earplugs

Many veteran rockers suffer from acute hearing damage because of the high sound levels they endured for years while playing music near loudspeakers or listening to music on loud headphones.

Recently, many of them began wearing special earplugs to protect their hearing during performances.

If an earplug decreases the sound level of the sound waves by 20 dB, what is the ratio of the final intensity If of the waves to their initial intensity Ii?

Example: Earplugs

  • Thus, the earplug reduces the sound intensity to 0.01 of the initial intensity.

Sources of Musical Sound

  • Musical sounds can be set up by oscillating strings, membranes, air columns, wooden blocks or steel bars, and many other oscillating bodies.

  • Recall again the standing waves, which can be set up on a stretched string that is fixed at both ends.

  • Waves traveling along the string are reflected back onto the string at each end. Then, the superposition of waves traveling in opposite directions produces a standing wave pattern (or oscillation mode).

  • Furthermore, the standing waves push back and forth against the surrounding air and thus generating a noticeable sound wave. This production of sound occurs in guitar, harp, violin, etc.

Sources of Musical Sound

  • Standing waves can also be set up in an air-filled pipe in a similar way. The reflection can occurs both when the end of the pipe is open or closed.

  • This production of sound is important in flute, oboe, pipe organ, etc.

  • Pipe organ

Sources of Musical Sound

Standing Waves in a Pipe

  • At closed end, air molecules cannot move  displacement at node, pressure at antinode

  • At open end, air molecules can move freely  displacement at antinode, pressure at node.

Sources of Musical Sound

  • The simplest standing wave pattern that can be set up in a pipe with two open ends is shown in the next figure.

  • There is an antinode across each open end. There is also a node across the middle of the pipe.

  • The standing wave pattern drawn here is called the fundamental mode or first harmonic.

  • For it to be set up, the sound waves in a pipe of length L must have a wavelength given by L = λ/2.

Two Open Ends

  • More generally, the resonant frequencies for a pipe of length L with two open ends corresponds to the wavelengths

  • Letting v be the speed of sound, we write the resonant frequencies for a pipe with two open ends as

  • Pipe, two open ends

  • Recall that n is called the harmonic number.

One Open End

  • The next figure shows some of the standing sound wave patterns that can be setup in a pipe with only one open end.

  • Across the open end there is an antinode and across the closed end there is a node.

  • More generally, the resonant frequencies for a pipe of length L with one open end corresponds to the wavelengths

  • The resonant frequencies for a pipe with one open end is then given by

  • Pipe, one open end

Musical Instruments

  • The length of a musical instrument reflects the range of frequencies over which the instrument is designed to function.

  • Smaller length implies higher frequencies.

Musical Instruments

  • In any oscillating system that give rise to a musical sound, whether it is a violin string or the air in an organ pipe, the fundamental and one or more of the higher harmonics are usually generated simultaneously.

  • We hear them together, superimposed as a net wave.

  • When different instruments are played at the same not, they produce the same fundamental frequency but different intensities for the higher harmonics.

  • Thus, because different instruments produce different net waves, they sound different to you even when they are played at the same note.

  • The waves produced by flute and oboe, at the same note


Pipe A, with length L, and pipe B, with length 2L, both have two open ends. Which harmonic of pipe B has the same frequency as the fundamental of pipe A?

  • Fundamental of pipe A

  • Second harmonic of pipe B

  • With same frequency as fundamental of pipe A

Example: Cardboard Tube

Weak background noises from a room set up the fundamental standing wave in a cardboard tube of length L = 67.0 cm with two open ends. Assume that the speed of sound in the air within the tube is 343 m/s

(a)What frequency do you hear from the tube?

  • Pipe, two open ends

(b)If you jam your ear against one end of the tube, what fundamental frequency do you hear from the tube?

  • Pipe, one open end

Homework 4: Two Speakers

Two speakers separated by distance d1 = 2 m are in phase. A listener observes at distance d2 = 3.75 m directly in front of one speaker. Consider the full audible range for normal human hearing, 20 Hz to 20 kHz. Sound velocity is 343 m/s.

What is the lowest frequency fmin,1 that gives minimum signal (destructive interference) at the listener’s ear?

What is the second lowest frequency fmin,2 that gives minimum signal?

What is the lowest frequency fmax,1 that gives maximum signal (constructive interference) at the listener’s ear?

What is the highest frequency fmax,n that gives maximum signal?

Homework 4A: Other Two Speakers

Two loudspeakers, A and B, are driven by the same amplifier and emit sinusoidal waves in phase. Speaker B is 12 m to the right of speaker A. The frequency of the waves emitted by each speaker is 686 Hz. Sound velocity is 343 m/s.

You are standing between the speakers, along the line connecting them, and are at a point of constructive interference.

How far must you walk toward speaker B to move to a point of destructive interference?

How far must you walk toward speaker B to move to another point of constructive interference?

Organ pipe C, with both ends open, has a fundamental frequency of 320 Hz. The fifth harmonic of organ pipe D, with one end open, has the same frequency as the third harmonic of pipe C. Determine the length of pipe C and pipe D if the speed of sound in air is 343 m/s.

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