In the coupled pendulums of Figure 4.3 let us write the...

In the coupled pendulums of Figure $4.3$ let us write the modulated frequency $\omega_{m}=\left(\omega_{2}-\omega_{1}\right) / 2$ and the average frequency $\omega_{a}=\left(\omega_{2}+\omega_{1}\right) / 2$ and assume that the spring is so weak that it stores a negligible amount of energy. Let the modulated amplitude $$ 2 a \cos \omega_{m} t \quad \text { or } \quad 2 a \sin \omega_{m} t $$ be constant over one cycle at the average frequency $\omega_{a}$ to show that the energies of the masses may be written $$ E_{x}=2 m a^{2} \omega_{a}^{2} \cos ^{2} \omega_{m} t $$ and $$ E_{y}=2 m a^{2} \omega_{a}^{2} \sin ^{2} \omega_{m} t $$ Show that the total energy $E$ remains constant and that the energy difference at any time is $$ E_{x}-E_{y}=E \cos \left(\omega_{2}-\omega_{1}\right) t $$ Prove that $$ E_{x}=\frac{E}{2}\left[1+\cos \left(\omega_{2}-\omega_{1}\right) t\right] $$ and $$ E_{y}=\frac{E}{2}\left[1-\cos \left(\omega_{2}-\omega_{1}\right) t\right] $$ to show that the constant total energy is completely exchanged between the two pendulums at the beat frequency $\left(\omega_{2}-\omega_{1}\right)$.

In the coupled pendulums of Figure $4.3$ let us write the modulated frequency $\omega_{m}=\left(\omega_{2}-\omega_{1}\right) / 2$ and the average frequency $\omega_{a}=\left(\omega_{2}+\omega_{1}\right) / 2$ and assume that the spring is so weak that it stores a negligible amount of energy. Let the modulated amplitude
$$
2 a \cos \omega_{m} t \quad \text { or } \quad 2 a \sin \omega_{m} t
$$
be constant over one cycle at the average frequency $\omega_{a}$ to show that the energies of the masses may be written
$$
E_{x}=2 m a^{2} \omega_{a}^{2} \cos ^{2} \omega_{m} t
$$
and
$$
E_{y}=2 m a^{2} \omega_{a}^{2} \sin ^{2} \omega_{m} t
$$
Show that the total energy $E$ remains constant and that the energy difference at any time is
$$
E_{x}-E_{y}=E \cos \left(\omega_{2}-\omega_{1}\right) t
$$
Prove that
$$
E_{x}=\frac{E}{2}\left[1+\cos \left(\omega_{2}-\omega_{1}\right) t\right]
$$
and
$$
E_{y}=\frac{E}{2}\left[1-\cos \left(\omega_{2}-\omega_{1}\right) t\right]
$$
to show that the constant total energy is completely exchanged between the two pendulums at the beat frequency $\left(\omega_{2}-\omega_{1}\right)$.

Key Concepts

Coupled Oscillators

Coupled oscillators are systems in which two or more oscillating bodies interact with each other through a coupling mechanism, such that the motion of one affects the motion of the others. This interaction leads to phenomena like energy transfer and the appearance of new patterns of oscillation, known as normal modes, where the system oscillates in a coordinated manner.

Normal Modes

Normal modes are the distinct and independent patterns of oscillation that a coupled system can exhibit. Each mode oscillates at its own characteristic frequency, and in a system with two degrees of freedom, there are typically two normal mode frequencies. Analyzing these modes helps in understanding how energy is distributed and exchanged within the system.

Beat Frequency

Beat frequency is the phenomenon observed when two oscillations with slightly different frequencies interfere with each other. The resulting motion is a rapid oscillation at an average frequency modulated by a slower amplitude envelope that oscillates at the beat frequency. This concept is crucial to explaining how energy is periodically exchanged between the oscillators.

Amplitude Modulation

Amplitude modulation in the context of coupled oscillators refers to the variation in the amplitude of the oscillation at a primary or carrier frequency by a lower frequency envelope. This modulation describes how the energy of the system oscillates between different parts of the system, manifesting as periodic increases and decreases in the energy of each oscillator.

Energy Conservation in Oscillatory Systems

In isolated or ideal oscillatory systems, the total energy is conserved despite the apparent transfer of energy between different components or modes. In the case of coupled oscillators, while the energy associated with individual oscillators may vary over time due to energy exchange, the sum of their energies remains constant, illustrating the principle of energy conservation.

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