|Example 1.9 The sound at a particular point consists of a main loudspeaker signal and a reflection at the same amplitude that has been delayed by 1 millisecond. What is the pressure amplitude at this point at 250 Hz, 500 Hz and 1 kHz?
The equation for pressure at a point due to a single frequency is given by the equation:
Pat a point = Psound amplitudesin(2πft) or Psound amplitudesin(360°ft)
where f = the frequency (in Hz)
and t = the time (in s)
Note the multiplier of 2π, or 360°, within the sine function is required to express accurately the position of the wave within the cycle. Because a complete rotation occurs every cycle, one cycle corresponds to a rotation of 360 degrees, or, more usually, 2π radians. This representation of frequency is called angular frequency (1 Hz (cycle per second) = 2π radians per second.
The effect of the delay to the difference in path lengths alters the time of arrival of one of the waves, and so the pressure at a point tdue to a single frequency delayed by some time, τ, is given by the equation:
Pat a point = Psound amplitudesin(2πf(t + τ))
or Psound amplitudesin(360°f(t + τ))
where τ = the delay (in s)
Add the delayed and undelayed sine waves together to give:
Ptotal = Pdelayedsin(360°f(t + τ)) + Pundelayedsin(360°ft)
Assuming that the delayed and undelayed signals are the same amplitude this can be re-expressed as:
Ptotal = 2P cos(360°f(τ/2))sin(360°f(t + τ/2))
The cosine term in this equation is determined by the delay and frequency and the sine term represents the original wave slightly delayed. Thus we can express the combined pressure amplitude of the two waves as:
Ptotal = 2P cos(360°f(τ/2))
Using the above equation we can calculate the effect of the delay on the pressure amplitude at the three different frequencies as:
Ptotal 250 Hz = 2P cos(360°f(τ/2))
= 2P cos(360° x 250 Hz x (1 x 10-3s/2)) = 1.41P
Ptotal 500 Hz = 2P cos(360°f(τ/2))
= 2P cos(360° x 500 Hz x (1 x 10-3s/2)) = 0
Ptotal 1 kHz = 2P cos(360°f(τ/2))
= 2P cos(360° x 1 kHz x (1 x 10-3s/2)) = 2P
These calculations show that the summation of correlated sources can be strongly frequency dependent and can vary between zero and twice the wave pressure amplitude.