Oscillating systems. Description and analysis - Building Acoustics - page 6

our physical measurement variables will always be functions of the positive frequencies.
Note the symmetry in the Equations (1.6) and (1.7). The relationship to the coefficients
used above will be

1

( )

X = a − j b ,

k k k

2

(1.8)

1

( )

X = ab + j .

− k k k

2

Hence

1 1

2 2

X = XX ⋅ = a b + = c

k k − k k k k

2 2

(1.9)

⎛⎞

b
a

k

and θ = arctg .

⎜⎟
⎝⎠

k

k

1.4.1.1 Energy in a periodic oscillation. Mean square and RMS-values

In sound or vibration measurements the function x ( t ) may, to mention a few examples,
represent the pressure in a sound wave at a given position in a room; the displacement,
the velocity or acceleration amplitude at a point on a wall or at a point on the surface of a
machine. Using the first example we put x ( t ) = p ( t ), where p is the pressure in the sound
wave with the unit Pa (Pascal). The instantaneous energy transported per unit time and
per unit area normal to the direction of propagation, i.e. the intensity of the wave, will be
proportional to the square of the pressure. By using Equation (1.3) or (1.5) , where we
now put a 0 (or c 0 ) equal to zero, we obtain

T

∞ ∞

1 1 1

( )

∑ ∑

∫

()

2 2 2 2 2 2

pt () = xt () = x t t d = a b + = c

, (1.10)

k k k

T 2 2

k = 1 k = 1

0

where the line above the pressure squared denotes taking the mean value over time. The
total energy flow is then given by the squared sum of the Fourier amplitudes.
Considering the total sound pressure as being a sum of sound pressure components, each
component associated with a given amplitude and frequency, it is reasonable to believe
that each component carries its part of the energy, i.e. that the energy in the k th harmonic
is proportional to c k 2 . However, we cannot claim this on the basis of Equation (1.10)
only.
We may use a similar argument when x ( t ) represents a velocity amplitude u ( t ) of a
part of a mechanical system, e.g. a vibrating plate. The mean square value of u( t ) will
then be proportional to the kinetic energy of this part of the system. However, in practical
measurement technique another quantity is more commonly used than the mean square
value. This is obtained by taking the root of it and then we arrive at the RMS-value (root-
mean-square-value) of the chosen variable. We then have

1 1

∞ ∞

⎡ ⎤ ⎡ ⎤

2 2

1

∑∑

2

( )

2 2

x = x () t = c = c , (1.11)

⎢ ⎥ ⎢ ⎥

rms k rms

k

2

⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦

k = 1 k = 1