Building Acoustics

Oscillating systems 9

1.4.2.1 Energy in transient motion

Again letting x(t) represent the time history of the sound pressure in air caused by a gun
shot, an explosion or similar events, the total energy represented by the integral

(^) ∫x^2 ()dtt
must be finite. From Equations (1.13) and (1.14) it is easy to show that

22

0

x()dtt | ( )|dXf f2 | ( )| dXf f^2

+∞ +∞ +∞

−∞ −∞

∫∫ ∫==. (1.16)

In other words; we may find the total energy either by an integration of the time function
or by an integration of the Fourier transform in the frequency domain. This is the reason
why the squared modulus |X(f)|^2 = X(f)⋅X(f) is called the energy spectral density, where
X(f) is the complex conjugate of X(f). The last form of the integral in Equation (1.16) is
possible because the time function x(t), representing all types of oscillatory motion, will
be a real function and therefore X(-f) = X*(f).

1.4.2.2 Examples of Fourier transforms

In practice we certainly must, in the first place, put a finite limit on the time T when
using our Fourier transform. Second, in measurement as well in calculations, the
transform is used in a discrete form (DFT). In this section we shall, however, show some
examples where the transform may readily be calculated analytically using Equation
(1.13). In this way we may vary the parameters to illustrate some important relationships
between the time and frequency domain representations.

A) A function describing a simple pulse of rectangular shape may be expressed as

x(t) = A for -T/2 ≤ t ≤ T/2
and x(t) = 0 otherwise.

Inserting this into the expression for X(f) (see Equation (1.13)) we obtain

sin( )

().

fT

Xf AT

fT

π

π

=

Figures 1.5 and 1.6 show some examples of the time function and the resulting modulus
|X(f)| of the transform. The amplitude A is arbitrarily set equal to 100 and the time T is
chosen 1, 5 and 10 ms, respectively. It should be noted that the spectrum broadens out
with decreasing pulse duration. However, as the amplitude A is constant the spectral
amplitudes must decrease. (Try to explain why.) An “infinitely” short pulse, represented
by the so-called Dirac δ-function, gives an infinite broad spectrum of constant amplitude,
a white spectrum. (Could you state the frequency amplitude of such a function?)