Question

Challenge Problem Beats When two sinusoidal waves travel through the same medium, a third wave is formed that is the sum of the two original waves. If the two waves have slightly different frequencies, the sum of the waves results in an interference pattern known as a beat. Musicians use this idea when tuning an instrument with the aid of a tuning fork. If the instrument and the tuning fork play the same frequency, no beat is heard. Suppose two waves given by the functions, \(y_{1}=3 \cos \left(\omega_{1} t\right)\) and \(y_{2}=3 \cos \left(\omega_{2} t\right)\) where \(\omega_{1}>\omega_{2}\) pass through the same medium, and each has a maximum at \(t=0\) sec. (a) How long does it take the sum function \(y_{3}=y_{1}+y_{2}\) to equal 0 for the first time? (b) If the periods of the two functions \(y_{1}\) and \(y_{2}\) are \(T_{1}=19 \mathrm{sec}\) and \(T_{2}=20 \mathrm{sec},\) respectively, find the first time the \(\operatorname{sum} y_{3}=y_{1}+y_{2}=0\) (c) Use the values from part (b) to graph \(y_{3}\) over the interval \(0 \leq x \leq 600 .\) Do the waves appear to be in tune?

Short answer

The first time the sum function equals zero is at 190 seconds. Use this to graph the function over the interval 0 to 600 seconds. The graph will show if the waves are in tune.

Step-by-step solution

Write the Sum Function

Given two waves, the sum function is written as \[ y_3 = y_1 + y_2 = 3 \cos(\omega_1 t) + 3 \cos(\omega_2 t) \]Using trigonometric identities, this can be simplified to\[ y_3 = 6 \cos \left( \frac{\omega_1 + \omega_2}{2} t \right) \cos \left( \frac{\omega_1 - \omega_2}{2} t \right) \]

Set the Sum Function to Zero

To find when the sum function first equals zero, set\[ y_3 = 6 \cos \left( \frac{\omega_1 + \omega_2}{2} t \right) \cos \left( \frac{\omega_1 - \omega_2}{2} t \right) = 0 \]This happens when either of the cosine factors is zero.

Solve for Time When Cosine is Zero

Next, solve the equation for the time, focusing on\[ \cos \left( \frac{\omega_1 - \omega_2}{2} t \right) = 0 \]The cosine function equals zero for the first time at\[ \frac{\omega_1 - \omega_2}{2} t = \frac{\pi}{2} \]Thus, solve for \( t \)\[ t = \frac{\pi}{2} \left( \frac{2}{\omega_1 - \omega_2} \right) = \frac{\pi}{\omega_1 - \omega_2} \]

Use Given Periods to Find Angular Frequencies

Use the periods \( T_1 \) and \( T_2 \) to find angular frequencies\[ \omega_1 = \frac{2\pi}{T_1} = \frac{2\pi}{19 \text{ sec}} \]\[ \omega_2 = \frac{2\pi}{T_2} = \frac{2\pi}{20 \text{ sec}} \]

Compute the Difference of Angular Frequencies

Compute the difference of angular frequencies\[ \omega_1 - \omega_2 = \frac{2\pi}{19} - \frac{2\pi}{20} \]Simplify the expression\[ \omega_1 - \omega_2 = 2\pi \left( \frac{1}{19} - \frac{1}{20} \right) = 2\pi \left( \frac{20 - 19}{19 \cdot 20} \right) = \frac{2\pi}{380} \]

Find First Zero of Sum Function Using Periods

Substitute the value in the expression for time\[ t = \frac{\pi}{\omega_1 - \omega_2} = \frac{\pi}{\frac{2\pi}{380}} = \frac{\pi \cdot 380}{2\pi} = 190 \text{ sec} \]Thus, the first zero for the sum function occurs at \( t = 190 \text{ sec} \).

Graph the Sum Function Over Given Interval

Using the calculated values and the given period, graph the function \( y_3 \) over the interval \[ 0 \leq t \leq 600 \text{ sec} \]Check if the waves appear to be in tune by observing the amplitude and frequency modulation in the graph.

Key concepts

sinusoidal waves

Sinusoidal waves are smooth, repetitive oscillations that can describe many natural and man-made phenomena. Their mathematical representations often involve trigonometric functions like sine and cosine. In this particular exercise, we deal with cosine waves represented as:

\(y_1 = 3 \, \cos \left( \omega_1 \, t \right)\) and \(y_2 = 3 \, \cos \left( \omega_2 \, t \right)\).

Here, \(\omega_1\) and \(\omega_2\) represent angular frequencies, and \(t\) denotes time. Sinusoidal waves are crucial in understanding periodic motion and wave interference, leading us to the formation of beats. Beats are essentially the result of superposing two sinusoidal waves with slightly different frequencies, resulting in a new wave that exhibits oscillations in amplitude.

trigonometric identities

Trigonometric identities are formulas that involve trigonometric functions and are true for any value of the variables. They are essential in solving trigonometric equations and simplifying expressions.

In this exercise, we use the sum-to-product identities to simplify the sum of two cosines:

\[ y_3 = y_1 + y_2 = 3 \, \cos (\omega_1 \, t) + 3 \, \cos (\omega_2 \, t) \]

The identity we employ is:

\[ \cos(A) + \cos(B) = 2 \, \cos \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right) \]

Applying this identity, we get:

\[ y_3 = 6 \, \cos \left( \frac{\omega_1 + \omega_2}{2} \, t \right) \cos \left( \frac{\omega_1 - \omega_2}{2} \, t \right) \]

This simplification is key to finding when the sum function equals zero.

angular frequencies

Angular frequency is a measure of how many oscillations or cycles a wave completes in a unit of time. It is denoted by \(\omega\) and is related to the period \(T\) of the wave through the equation:

\[ \omega = \frac{2\pi}{T} \]

Given the periods of the waves in the exercise, \( T_1 = 19 \, \text{sec} \) and \( T_2 = 20 \, \text{sec} \), we calculate their angular frequencies as follows:

\[ \omega_1 = \frac{2\pi}{T_1} = \frac{2\pi}{19} \]

\[ \omega_2 = \frac{2\pi}{T_2} = \frac{2\pi}{20} \]

The difference in these angular frequencies determines the rate at which beats are formed.

graphing functions

Graphing functions allows us to visualize periodic patterns and interference effects, such as beats. For the combined wave \( y_3 \), after determining the first zero at \( t = 190 \, \text{sec} \), we graph the function over \( 0 \leq t \leq 600 \) \(\text{sec} \).

Graphing helps in observing whether the waves are in tune. In this case, look for consistent patterns or irregularities in the amplitude modulation. If the waves were perfectly in tune, the interference pattern would be uniform without any amplitude variation. However, due to the slight frequency mismatch, the resulting beat pattern will display alternating loud and soft points, indicating periodic amplitude changes due to interference.