Sources of Musical Sound — Question 6

Question 6

Pressure and velocity microphones see complementary pipe modes

An ideal open-open tube has pressure modes $p_n=P_n\sin(n\pi x/L)\cos\omega_nt$. A pressure sensor and a nonloading axial-velocity sensor are both placed at the midpoint. Use $\rho_0u_t=-p_x$ and $\omega_n=cn\pi/L$.

1. Derive the velocity mode shape.

2. Which of modes $n=1$ through $6$ can each sensor detect at the midpoint? Explain why a missing pressure peak does not prove that the mode is absent.

Question 6 - Solution

Pressure and velocity microphones see complementary pipe modes

(a) Method and calculation

Differentiating pressure gives$$p_x=P_n\frac{n\pi}{L}\cos(n\pi x/L)\cos\omega_nt.$$Integrating the momentum equation in time, with no steady flow, gives$$u_n=-\frac{P_n}{\rho_0c}\cos(n\pi x/L)\sin\omega_nt.$$

(b) Interpretation and further calculation

At $x=L/2$, the pressure factor is $\sin(n\pi/2)$, nonzero for $n=1,3,5$. The velocity factor is $\cos(n\pi/2)$, nonzero for $n=2,4,6$. Even modes can store energy and move fluid at the midpoint while producing zero pressure there. Sensor coupling, not just allowed eigenfrequency, determines visibility.

Physical check

The two sensor sets are complementary here because the pressure and velocity spatial factors differ by a quarter wavelength.