This set of 32 problems targets your ability to determine wave quantities such as frequency, period, wavelength, intensity and speed from verbal descriptions and diagrams of physical situations pertaining to sound waves and resonance in strings and air columns. Problems range in difficulty from the very easy and straight-forward to the very difficult and complex. The more difficult problems are color-coded as *blue problems*.

The speed of any wave (v) is defined as the distance traveled (d) per time of travel (t) and is described by the following equation:

v = d / t

Thus, the distance traveled by a wave is related to the time required for it to travel that distance.

The speed of sound, like the speed of any wave, is dependent upon the properties of the medium through which it is moving. For a sound wave moving through air, the primary property of air which effects its speed is the temperature of the air. There are a couple of equations which describe the speed (v) - temperature (T) relationship; the following might be the easiest to remember:

v = 331 m/s + (0.6 m/s/°C) •T

where **T** is the Celsius temperature of the air through which the sound wave is moving. At 0°C, the speed of sound is 331 m/s. For every degree Celsius above 0°C, the speed of sound increases by approximately 0.6 m/s. This equation provides a rather accurate estimate of the speed of sound for temperatures upwards towards 50°C.

Sound waves are produced by a vibrating object - vocal chords, guitar string, or diaphragm on a speaker. The sound waves begin at a point (or approximately a point) and then propagate through space in three dimensions. As the sound wave propagates, it creates a wave front that fills the surface area of an ever-expanding sphere. A sound wave (like any wave) is often referred to as an energy-transport phenomenon. The rate at which energy is put into the wave is referred to as the **power of the source**. Power (P) is expressed in units of Watts. As these vibrations propagate through space, they become less intense as the size of the spherical wave front expands. Whatever energy is created by the wave at the source fills the surface area of a sphere some distance R away. Because the sphere is constantly expanding, the energy is becoming *diluted* with increasing distance from the source. The **sound intensity** (I) at any given location is defined as the rate at which energy passes through that location. Because the wave energy is spreading over a surface *area*, intensity is often expressed in units of Watts/meter^{2} and given by the equation

I = P / (4 • Π • R^{2})

The human ear is sensitive enough to detect the smallest of vibrations. The lowest amplitude vibration which most humans can hear is defined as the **threshold of hearing** (TOH). While such a sound will be different for different people, the intensity associated with this sound is defined as **1.0 x 10 ^{-12} W/m^{2}**. The range of sound intensities which a typical human can detect is enormous. Sound which 100 billion times more intense than the threshold of hearing (1 x 10

The deciBel level in decibel can be properly determined from the intensity (**I**) of a sound using the following equation:

dB = 10 • log ( I / 1.0 x 10^{-12} W/m^{2} )

Often times, the deciBel level is known and the intensity in W/m^{2} is desired. In such instances the equation below can be used.

I = 1.0 x 10^{-12} W/m^{2} • 10^{x} where x = dB/10

The speed of a wave depends upon the properties of the medium through which it is transmitted, and NOT upon the properties of the wave itself. For sound waves being transmitted through strings, wires, ropes and cables, the primary properties effecting wave speed are the tension of the medium and the mass density of the medium. Tension pertains to the force with which the two ends of the medium are pulled tight. Being a force, it is expressed in units of Newtons. Mass density pertains to the mass per unit length of the string, wire, rope or cable and is expressed in standard units of kilogram/meter (kg/m). The equation expressing the relationship between these variables is

v = √(T/μ)

where v represents the wave speed, T represents the tension, and μ represents the mass density. When using these equations, it is important to pay attention to the units with which the given quantities are expressed and to make appropriate conversions where necessary. It is recommended that substitutions be made into the equation using standard metric units. For speed, use m/s; for tension, use Newtons (abbreviated N); and for mass density, use kg/m.

Many of the questions in this problem set target your ability to analyze physical situations involving the wavelength-frequency-speed relationship for standing wave patterns in strings, wires, ropes, cables and air columns. Any wave, whether a standing wave or a traveling wave, will have a wavelength-frequency-speed relationship which follows the wave equation:

v = f • λ

where v represents the speed (or velocity) of the wave, f represents the frequency of the wave, and λ represents the wavelength of the wave. As mentioned above, the speed of a wave is dependent upon the properties of the medium. For strings, wires, ropes and cables, the properties of importance are the tension and mass density. For air columns, the property of importance is the temperature of the air. As such, the properties of the medium are also related to the frequency and the wavelength of the wave.

Waves introduced into a string, wire, rope or cable will typically travel the length of the medium and reflect back upon reaching its end. At certain frequencies, the reflected portion of the wave meets up with the original wave to create a pattern known as a **standing wave pattern**. In a standing wave pattern there are points along the medium which appear as if they are always standing still. These points are known as **nodes** and are easily remembered as the points of **no** **des**placement (properly spelled as *displacement*). Separating the nodes are **anti-nodes**: points of maximum positive and negative displacement. In such standing wave patterns, there is a unique half-number relationship between the length of the medium and the wavelength of the waves which have established the pattern seen. These relationships are shown below for the standing wave patterns having one anti-node (first harmonic), two anti- nodes (second harmonic) and three anti- nodes (third harmonic).

It is clear from the above graphic that the length of the string, wire, rope or cable is related to the wavelength of the standing wave which is established within it. As such, the length of the medium is related mathematically to the frequency of the wave and the speed of the waver (or the properties of the medium upon which wave speed depends).

A standing wave pattern such as those shown in the graphic above is established within a medium only when it is being disturbed at specific frequencies. Not any frequency will result in a standing wave pattern; only the discrete frequencies which lead to wavelength values which are mathematically related to the length of the medium as illustrated by the equations in the graphic above. Such frequencies are known as the harmonic frequencies of the string, wire, rope or cable. As observed in the graphic above, the pattern associated with the second harmonic has a wavelength which is one-half the wavelength of the pattern associated with the first harmonic. And the pattern associated with the third harmonic has a wavelength which is one-third the wavelength of the pattern associated with the first harmonic. Continuing this same logic, one would reason, that the pattern associated with the fifth harmonic has a wavelength which is one-fifth the wavelength of the pattern associated with the first harmonic. And in general, the pattern associated with the *n ^{th}* harmonic has a wavelength which is 1 /

f_{n} = n • f_{1}

where **f _{n}** is the frequency of any harmonic pattern,

Solving problems relating to resonating strings targets a student's ability to relate the frequency, wavelength and speed of waves to properties of the string and to the length of the string. The graphic below depicts the relationships between the various quantities in such problems. As is the usual case, when approaching a problem, first identify what you know and what you are trying to find. Locate the knowns and unknowns on the graphic below and plot out a strategy which allows you to determine the unknown quantity. The strategy for solving for the unknown will be centered around the relationships depicted in the graphic. The stated equations provide the mathematical expression of those relationships.

As just mentioned, a standing wave can be established in a string when vibrating at one of its resonance frequencies. Similarly, a column of air can resonate as well provided that another object vibrating at one of the resonance frequencies forces the air into vibration. The resonance vibrations of an air column is the basis of many of the later problems in this problem set. The air column is either open at both ends (open-end air column) or open at one end and closed at the other (closed-end air column). A resonance situation in an open-end air column is characterized by the presence of an anti-node at each of the open ends, creating the standing wave patterns shown below. Each pattern is referred to as a harmonic and has its own unique wavelength and frequency. As shown in the graphic, there is a distinct relationship between the wavelength of the standing wave and the length of the air column. Knowing the pattern allows one to relate the length to the wavelength and ultimately to the frequency and the speed.

A closed-end air column is open to the surrounding air at one end and closed at the other end. A resonance situation in a closed-end air column is characterized by the presence of an anti-node at the open end and a node at the closed end, creating the standing wave patterns shown below. Again, there is a distinct relationship between the length of the air columns and the wavelength for each of the harmonics. Knowing this length-wavelength relationship allows one to relate the length of an air column to the speed and the frequency at which the air inside naturally vibrates.

A strategy for solving problems related to resonating strings was discussed above. A similar strategy can be applied in the approach to problems involving resonating air columns. Such problems target a student's ability to relate the frequency, wavelength and speed of waves to properties of the air column (temperature) and to the length of the air column. The graphic below depicts the relationships between the various quantities in such problems. As is the usual case, when approaching a problem, first identify what you know and what you are trying to find. Locate the knowns and unknowns on the graphic below and plot out a strategy which allows you to determine the unknown quantity. The strategy for solving for the unknown will be centered around the relationships depicted in the graphic. The stated equations provide the mathematical expression of those relationships.

When two sound sources of very similar yet different frequencies meet at an observer's ears the phenomenon of **beats** is observed. This phenomenon is perceived by an observer as a sound which fluctuates in amplitude over the course of time. For instance, if two tuning forks - one with a frequency of 256 Hz and the other with a frequency of 254 Hz - produce sound waves, then an observer would hear a fluctuation in amplitude at a frequency of 2 Hz. This 2 Hz fluctuation in frequency is known as the beat frequency and is equivalent to the difference in frequency between the two sources. The fluctuations in amplitude which are observed is the result of the interference of the two waves. A low amplitude sound is observed when a compression from one source meets up with a rarefaction from the other source. A high amplitude sound is observed when a compression from one source meets up with a compression from the other source (or a rarefaction with a rarefaction).

An effective problem solver by habit approaches a physics problem in a manner that reflects a collection of disciplined habits. While not every effective problem solver employs the same approach, they all have habits which they share in common. These habits are described briefly here. An effective problem-solver...

- ...reads the problem carefully and develops a mental picture of the physical situation. If needed, they sketch a simple diagram of the physical situation to help visualize it.
- ...identifies the known and unknown quantities, often times recording them on the diagram iteself. They equate given values to the symbols used to represent the corresponding quantity (e.g., v = 345 m/s, λ = 1.28 m, f = ???).
- ...plots a strategy for solving for the unknown quantity; the strategy will typically center around the use of physics equations be heavily dependent upon an understanding of physics principles.
- ...identifies the appropriate formula(s) to use, often times writing them down. Where needed, they perform the needed conversion of quantities into the proper unit.
- ...performs substitutions and algebraic manipulations in order to solve for the unknown quantity.

The following pages from The Physics Classroom Tutorial may serve to be useful in assisting you in the understanding of the concepts and mathematics associated with these problems.

- Speed of Sound
- Intensity and the Decibel Scale
- Standing Wave Patterns
- Fundamental Frequency and Harmonics
- Physics of Guitar Strings
- Open-End Air Columns
- Closed-End Air Columns
- Interference and Beat Formation

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