AN APPLICATION TO SOUND
10. AN APPLICATION TO SOUND
We have said that when a sound wave passes through the air and we hear it, the air pressure where we are varies with time. Suppose the excess pressure above (and below) atmospheric pressure in a sound wave is given by the graph in Figure 10.1. (We shall not be concerned here with the units of p; however, reasonable units in Figure 10.1 would be p in 10 −6 atmospheres.) Let us ask what frequencies we hear when we listen to this sound. To find out, we expand p(t) in a Fourier series. The
period of p(t) is 1 262 ; that is, the sound wave repeats itself 262 times per second. We have called the period 2l in our formulas, so here l = 1 524 . The functions we have called sin(nπx/l) here become sin 524nπt. We can save some work by observing
Figure 10.1
that p(t) is an odd function; there are then only sine terms in its Fourier series and we need to compute only b n . Using (9.4), we have
b n = 2(524)
p(t) sin 524nπt dt
sin 524nπt dt − (1048)
sin 524nπt dt
From this we can compute the values of b n for the first few values of n:
b 8 = 0, etc. 4π
Section 10 An Application to Sound 373
Then we have
sin(524 · 3πt) (10.3)
1 30 sin(524 · 2πt)
p(t) =
sin(524 · 5πt)
30 sin(524 · 6πt)
sin(524 · 7πt)
5 6 7 We can see just by looking at the coefficients that the most important term
is the second one. The first term corresponds to the fundamental with frequency 262 vibrations per second (this is approximately middle C on a piano). But it is much weaker in this case than the first overtone (second harmonic) corresponding to the second term; this tone has frequency 524 vibrations per second (approximately high C). (You might like to use a computer to play one or several terms of the series.) The sixth harmonic (corresponding to n = 6) and also the harmonics for n = 10, 14, 18, 22, and 26 are all more prominent (that is, have larger coefficients) than the fundamental. We can be even more specific about the relative importance of the various frequencies. Recall that in discussing a simple harmonic oscillator, we showed that its average energy was proportional to the square of its velocity amplitude. It can be proved that the intensity of a sound wave (average energy striking unit area of your ear per second) is proportional to the average of the square of the excess pressure. Thus for a sinusoidal pressure variation A sin 2πf t,
the intensity is proportional to A 2 . In the Fourier series for p(t), the intensities of the various harmonics are then proportional to the squares of the corresponding Fourier coefficients. (The intensity corresponds roughly to the loudness of the tone—not exactly because the ear is not uniformly sensitive to all frequencies.) The relative intensities of the harmonics in our example are then:
2 3 4 5 6 7 8 9 10 · · · Relative intensity =1 225 1 1 1 9 1 0 25 25 49 0 81 9 ···
From this we see even more clearly that we would hear principally the second harmonic with frequency 524 (high C).
PROBLEMS, SECTION 10
In Problems 1 to 3, the graphs sketched represent one period of the excess pressure p(t) in a sound wave. Find the important harmonics and their relative intensities. Use a computer to play individual terms or a sum of several terms of the series.
374 Fourier Series and Transforms Chapter 7
In Problems 4 to 10, the sketches show several practical examples of electrical signals (voltages or currents). In each case we want to know the harmonic content of the signal, that is, what frequencies it contains and in what proportions. To find this, expand each function in an appropriate Fourier series. Assume in each case that the part of the graph shown is repeated sixty times per second.
4. Output of a simple d-c generator; the shape of the curve is the absolute value of a sine func- tion. Let the maximum voltage be 100 v.
5. Rectified half-wave; the curve is a sine function for half the cycle and zero for the other half. Let the maximum current be 5 amp. Hint: Be careful! The value of l here is 1/60, but I(t) = sin t only from t = 0 to t = 1/120.
6. Triangular wave; the graph consists of two straight lines whose equations you must write! The maximum voltage of 100 v occurs at the middle of the cycle.
7. Sawtooth 8. Rectified sawtooth
9. Square wave 10. Periodic ramp function
Section 11 Parseval’s Theorem 375
Parts
» CONVERGENCE TESTS FOR SERIES OF POSITIVE TERMS; ABSOLUTE CONVERGENCE
» CONDITIONALLY CONVERGENT SERIES
» POWER SERIES; INTERVAL OF CONVERGENCE
» EXPANDING FUNCTIONS IN POWER SERIES
» TECHNIQUES FOR OBTAINING POWER SERIES EXPANSIONS
» ACCURACY OF SERIES APPROXIMATIONS
» COMPLEX POWER SERIES; DISK OF CONVERGENCE
» ELEMENTARY FUNCTIONS OF COMPLEX NUMBERS
» POWERS AND ROOTS OF COMPLEX NUMBERS
» THE EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS
» LINEAR COMBINATIONS, LINEAR FUNCTIONS, LINEAR OPERATORS
» LINEAR DEPENDENCE AND INDEPENDENCE
» SPECIAL MATRICES AND FORMULAS
» EIGENVALUES AND EIGENVECTORS; DIAGONALIZING MATRICES
» APPLICATIONS OF DIAGONALIZATION
» A BRIEF INTRODUCTION TO GROUPS
» POWER SERIES IN TWO VARIABLES
» APPROXIMATIONS USING DIFFERENTIALS
» CHAIN RULE OR DIFFERENTIATING A FUNCTION OF A FUNCTION
» APPLICATION OF PARTIAL DIFFERENTIATION TO MAXIMUM AND MINIMUM PROBLEMS
» MAXIMUM AND MINIMUM PROBLEMS WITH CONSTRAINTS; LAGRANGE MULTIPLIERS
» ENDPOINT OR BOUNDARY POINT PROBLEMS
» DIFFERENTIATION OF INTEGRALS; LEIBNIZ’ RULE
» APPLICATIONS OF INTEGRATION; SINGLE AND MULTIPLE INTEGRALS
» CHANGE OF VARIABLES IN INTEGRALS; JACOBIANS
» APPLICATIONS OF VECTOR MULTIPLICATION
» DIRECTIONAL DERIVATIVE; GRADIENT
» SOME OTHER EXPRESSIONS INVOLVING ∇
» GREEN’S THEOREM IN THE PLANE
» THE DIVERGENCE AND THE DIVERGENCE THEOREM
» THE CURL AND STOKES’ THEOREM
» SIMPLE HARMONIC MOTION AND WAVE MOTION; PERIODIC FUNCTIONS
» APPLICATIONS OF FOURIER SERIES
» COMPLEX FORM OF FOURIER SERIES
» LINEAR FIRST-ORDER EQUATIONS
» OTHER METHODS FOR FIRST-ORDER EQUATIONS
» SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS AND ZERO RIGHT-HAND SIDE
» SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS AND RIGHT-HAND SIDE NOT ZERO
» OTHER SECOND-ORDER EQUATIONS
» SOLUTION OF DIFFERENTIAL EQUATIONS BY LAPLACE TRANSFORMS
» A BRIEF INTRODUCTION TO GREEN FUNCTIONS
» THE BRACHISTOCHRONE PROBLEM; CYCLOIDS
» SEVERAL DEPENDENT VARIABLES; LAGRANGE’S EQUATIONS
» TENSOR NOTATION AND OPERATIONS
» KRONECKER DELTA AND LEVI-CIVITA SYMBOL
» PSEUDOVECTORS AND PSEUDOTENSORS
» VECTOR OPERATORS IN ORTHOGONAL CURVILINEAR COORDINATES
» ELLIPTIC INTEGRALS AND FUNCTIONS
» GENERATING FUNCTION FOR LEGENDRE POLYNOMIALS
» COMPLETE SETS OF ORTHOGONAL FUNCTIONS
» NORMALIZATION OF THE LEGENDRE POLYNOMIALS
» THE ASSOCIATED LEGENDRE FUNCTIONS
» GENERALIZED POWER SERIES OR THE METHOD OF FROBENIUS
» THE SECOND SOLUTION OF BESSEL’S EQUATION
» OTHER KINDS OF BESSEL FUNCTIONS
» ORTHOGONALITY OF BESSEL FUNCTIONS
» SERIES SOLUTIONS; FUCHS’S THEOREM
» HERMITE FUNCTIONS; LAGUERRE FUNCTIONS; LADDER OPERATORS
» LAPLACE’S EQUATION; STEADY-STATE TEMPERATURE IN A RECTANGULAR PLATE
» THE DIFFUSION OR HEAT FLOW EQUATION; THE SCHR ¨ODINGER EQUATION
» THE WAVE EQUATION; THE VIBRATING STRING
» STEADY-STATE TEMPERATURE IN A CYLINDER
» VIBRATION OF A CIRCULAR MEMBRANE
» STEADY-STATE TEMPERATURE IN A SPHERE
» INTEGRAL TRANSFORM SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS
» EVALUATION OF DEFINITE INTEGRALS BY USE OF THE RESIDUE THEOREM
» THE POINT AT INFINITY; RESIDUES AT INFINITY
» SOME APPLICATIONS OF CONFORMAL MAPPING
» THE NORMAL OR GAUSSIAN DISTRIBUTION
» STATISTICS AND EXPERIMENTAL MEASUREMENTS
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