$3.4.19.$ The graph of coordinate versus time for a motion that is the sum of two harmonic oscillations is shown in the figure. Use it to determine the amplitudes and frequencies of these oscillations.
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### Solution
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<b>1. Extracting Amplitudes from the Graphical Envelope</b>
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<b>1. Extracting Amplitudes from the Graphical Envelope</b>\
The given graph illustrates a classic beating phenomenon resulting from the linear superposition of two harmonic oscillations with nearly equal amplitudes and slightly different frequencies ($\omega_1\approx\omega_2$).
The general equation describing the displacement $x(t)$ of such a combined system is:
$$x(t)=a_1\sin(\omega_1t)+a_2\sin(\omega_2t)$$
Looking closely at the graph parameters:
— <b>Maximum Envelope Peak ($A$):</b> This occurs when the two individual oscillations interfere constructively:
$$A_{\max}=a_1+a_2=A$$
— <b>Minimum Envelope Neck ($B$):</b> This occurs when the two components interfere destructively, opposing each other:
$$A_{\min}=|a_1-a_2|=B$$
@@ -24,15 +20,15 @@Solution
We can solve this simple algebraic system to isolate the distinct amplitudes $a_1$ and $a_2$ (assuming $a_1>a_2$):
$$a_1=\frac{A+B}{2}, \quad a_2=\frac{A-B}{2}$$
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<b>2. Extracting Frequencies from the Graphical Periods</b>
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<b>2. Extracting Frequencies from the Graphical Periods</b>\
The graph defines two specific time intervals on the horizontal axis:
— $\tau$<b>(Period of Fast Oscillations):</b> This corresponds to the time it takes to complete one full cycle of the rapid inner wave. It is directly tied to the average carrier frequency:
— $T$<b>(Beat Period):</b> This represents the time interval between two consecutive minimums (necks) of the slow amplitude envelope. The beat frequency is related to the difference between the two frequencies:
$3.4.19.$ The graph of coordinate versus time for a motion that is the sum of two harmonic oscillations is shown in the figure. Use it to determine the amplitudes and frequencies of these oscillations.
$3.4.19.$ The graph of coordinate versus time for a motion that is the sum of two harmonic oscillations is shown in the figure. Use it to determine the amplitudes and frequencies of these oscillations.
### Solution
### Solution
<b>1. Extracting Amplitudes from the Graphical Envelope</b>
<b>1. Extracting Amplitudes from the Graphical Envelope</b>\
The given graph illustrates a classic beating phenomenon resulting from the linear superposition of two harmonic oscillations with nearly equal amplitudes and slightly different frequencies ($\omega_1\approx\omega_2$).
The given graph illustrates a classic beating phenomenon resulting from the linear superposition of two harmonic oscillations with nearly equal amplitudes and slightly different frequencies ($\omega_1\approx\omega_2$).
The general equation describing the displacement $x(t)$ of such a combined system is:
The general equation describing the displacement $x(t)$ of such a combined system is:
$$x(t)=a_1\sin(\omega_1t)+a_2\sin(\omega_2t)$$
$$x(t)=a_1\sin(\omega_1t)+a_2\sin(\omega_2t)$$
Looking closely at the graph parameters:
Looking closely at the graph parameters:
— <b>Maximum Envelope Peak ($A$):</b> This occurs when the two individual oscillations interfere constructively:
— <b>Maximum Envelope Peak ($A$):</b> This occurs when the two individual oscillations interfere constructively:
$$A_{\max}=a_1+a_2=A$$
$$A_{\max}=a_1+a_2=A$$
— <b>Minimum Envelope Neck ($B$):</b> This occurs when the two components interfere destructively, opposing each other:
— <b>Minimum Envelope Neck ($B$):</b> This occurs when the two components interfere destructively, opposing each other:
$$A_{\min}=|a_1-a_2|=B$$
$$A_{\min}=|a_1-a_2|=B$$
@@ -24,15 +20,15 @@Solution
We can solve this simple algebraic system to isolate the distinct amplitudes $a_1$ and $a_2$ (assuming $a_1>a_2$):
We can solve this simple algebraic system to isolate the distinct amplitudes $a_1$ and $a_2$ (assuming $a_1>a_2$):
$$a_1=\frac{A+B}{2}, \quad a_2=\frac{A-B}{2}$$
$$a_1=\frac{A+B}{2}, \quad a_2=\frac{A-B}{2}$$
<b>2. Extracting Frequencies from the Graphical Periods</b>
<b>2. Extracting Frequencies from the Graphical Periods</b>\
The graph defines two specific time intervals on the horizontal axis:
The graph defines two specific time intervals on the horizontal axis:
— $\tau$<b>(Period of Fast Oscillations):</b> This corresponds to the time it takes to complete one full cycle of the rapid inner wave. It is directly tied to the average carrier frequency:
— $\tau$<b>(Period of Fast Oscillations):</b> This corresponds to the time it takes to complete one full cycle of the rapid inner wave. It is directly tied to the average carrier frequency:
— $T$<b>(Beat Period):</b> This represents the time interval between two consecutive minimums (necks) of the slow amplitude envelope. The beat frequency is related to the difference between the two frequencies:
— $T$<b>(Beat Period):</b> This represents the time interval between two consecutive minimums (necks) of the slow amplitude envelope. The beat frequency is related to the difference between the two frequencies:
