Engineering Acoustics. Malcolm J. Crocker

(see Chapter 2 of this book) is given by

      where A = (Mω_d)⁻¹, α = R/2 M and λ = ω_d is known as the damped “natural” angular frequency. Find the Fourier spectrum representation of this impulse response.

      Solution

      Using the mathematical property e^jθ = cos θ + j sin θ, we can write

. Then, Eq. (1.6) is

The impulse response and its Fourier spectrum are shown in Figure 1.7. We notice that replacing α and λ by the corresponding values in terms of the stiffness K, mass M, and damping constant R, of the damped mass‐spring system, the Fourier spectrum becomes (compare with Eq. (2.18))

Figure 1.7 Time and frequency domain representations of the transient response of the impulse response of a damped vibration of a mass‐spring system.

      1.3.4 Mean Square Values

      In the case of the pure tone a useful quantity to determine is the mean square value, i.e. the time average of the signal squared 〈x²(t)〉_t [8]

(1.7)

      where 〈〉_t denotes a time average.

      For the pure tone in Figure 1.2a then we obtain

(1.8)

      where A is the signal amplitude.

      The root mean square value is given by the square root of 〈x²(t)〉_t or

(1.9)

      For the general case of the complex pure tone in Eq. (1.1) or (1.2) we obtain:

(1.10)

      or

      (1.11)

      since . The mean square value then is the sum of the squares of all the harmonic components of the wave weighted by a constant of 1/2.

      Example 1.4

      Determine the mean square and rms values of the signal in Figure 1.3.

      Solution

We can use Eq. (1.7) to determine its mean square value,

The same result is obtained from its Fourier series representation using Eq. (1.10):

Recalling that the root mean square value is given by the square root of the mean square value, the rms value of this saw tooth signal is . Note the difference between the rms value obtained in this example and that in Eq. (1.9) for a sine wave.

      1.3.5 Energy and Power Spectral Densities

      In the case of nonperiodic signals (see Section 1.3.2), a quantity called the energy density function or equivalently the energy spectral density, S(f), is defined:

(1.12)

The energy spectral density S(f) is the “energy” of the sound or vibration signal in a bandwidth of 1 Hz. Note that S(ω) = 2πS(f) where S(ω) is the “energy” in a 1 rad/s bandwidth. We use the term “energy” because if x(t) were converted into a voltage signal, S(ω) would have the units of energy if the voltage were applied across a 1 Ω resistor. In the case of the pure tone, if x(t) is assumed to be a voltage, then the mean square value in Eq. (1.8) represents the power in watts.

      In the case of random sound or vibration signal we define a power spectral density G_x(f). This may be derived through the filtering – squaring – averaging approach or the finite Fourier transform approach. We will consider both approaches in turn.

      Suppose we filter the time signal through a filter of bandwidth Δf, then the mean square value

      (1.13)

      where x(t,f,Δf) is the filtered frequency component of the signal after it is passed through a filter of bandwidth Δf centered on frequency f. In the practical case, the filter bandwidth, Δf,