Question

A seated musician plays a D*5 note at 622 Hz. How much time At does it take for 421 air pressure maxima to pass a stationary listener? At = S You would like to express the air pressure oscillations at a point in space in the given form. P(t) = Pmaxcos (Bt) If t is measured in seconds, what value should the quantity B have? B= If t is measured in seconds, what units should the quantity B have?

Answer 1

f = 622 H2 pe 421 ts ala 421 622 10.6768 see. x € = A Coswt H+) A Cas Caft) PG) - Pmare Cos CBt) B = 2 Tf B = 2a (622) B 3908,14 Hz

Answer 2

The note that the musician plays is a sound wave, which is a longitudinal traveling wave. At a fixed point in space, this traveling wave becomes just an oscillation in the local air pressure. The time between one pressure maximum and the next is called the period and is the inverse of the frequency.

$$\begin{array}{rl}T& =\frac{1}{f}\\ & =\frac{1}{932\text{ Hz}}\\ & =0.00107\text{ s}\end{array}$$

The time that it takes for 796$796$ air pressure maxima to pass a point in space is 796$796$ multiplied by the period.

$$\begin{array}{rl}\mathrm{\Delta }t& =\left(796\right)T\\ & =\frac{796}{f}\\ & =\frac{796}{932\text{ Hz}}\\ & =0.854\text{ s}\end{array}$$

To express the air pressure oscillations in the given form, the quantity ?$B$ must be the angular frequency, ?$\omega$.

$$P\left(t\right)=P_{\text{max}}\cos \left(\omega t\right)$$

This is because the argument of the cosine function must be in angular units. The air pressure oscillates once every period, whereas the cosine function repeats every 2?$2\pi$ radians. Thus, the product of ?$\omega$ and the period should be equal to 2?$2\pi$ radians.

$$\omega T=2\pi \text{ radians}$$

Solving this relationship for ?$\omega$ gives the definition of angular frequency, which can then be found from either the period or the frequency.

$$\begin{array}{rl}\omega & =\frac{2\pi \text{ radians}}{T}\\ & =\left(2\pi \text{ radians}\right)f\\ & =\left(2\pi \text{ radians}\right)\times \left(932\text{ Hz}\right)\\ & =5860\frac{\text{radians}}{\text{s}}\end{array}$$

Therefore the value of ?$B$ is 5860 and it has units of radians per second.