Oscillations and Waves

1. Equation of Harmonic Oscillations

Basic Time Dependencies:

x (t) = X_{m} \cdot \sin (ω t + φ)

v (t) = ω \cdot X_{m} \cdot \cos (ω t + φ)

a (t) = - ω^{2} \cdot X_{m} \cdot \sin (ω t + φ)

What do these formulas describe?

These equations describe the parameters of harmonic oscillations:

x(t) — displacement (coordinate relative to equilibrium position);
v(t) — velocity (first derivative with respect to time);
a(t) — acceleration (second derivative, opposite to displacement).

All three functions depend on:
• amplitude xₘ (maximum deviation),
• angular frequency ω (rate of phase change),
• initial phase φ (time shift of the beginning of oscillation).

Acceleration is always proportional to displacement, but opposite in sign: maximum in magnitude at extreme points, zero at equilibrium.

2. Period and Frequency of Oscillations

Relationship between T, f, and ω:

T = \frac{1}{f}

ω = 2 π \cdot f

What do these formulas describe?

T — period, the time of one complete oscillation (in seconds).
f — frequency, the number of oscillations per second (in Hz): f = 1 / T
ω — angular frequency, defines the phase velocity: ω = 2πf

These quantities relate the temporal characteristics of motion to the equations x(t), v(t), a(t).

3. Periods of Simple Oscillatory Systems

Period of Spring System Oscillations:

T = 2 π \cdot \sqrt{\frac{m}{k}}

What does this formula describe?

The period of a spring pendulum depends on the mass m and the spring constant k. The heavier the load — the longer one oscillation, the stiffer the spring — the faster.

Period of a Mathematical Pendulum (small oscillations):

T = 2 π \cdot \sqrt{\frac{l}{g}}

What does this formula describe?

The period of a swinging weight on a thread depends only on the length of the thread l and the acceleration due to gravity g. The mass of the weight does not affect it. The formula is applicable for small angles of deviation (up to ~10°).

4. Mechanical Wave

Equation for the relationship between velocity, frequency, and wavelength:

v = λ \cdot f

What does this formula describe?

A wave transmits oscillations through a medium at a certain velocity v.
λ — wavelength: the distance between successive crests/troughs or identical phases of oscillations.
f — frequency: how many oscillations occur per second.

This formula is universal: it applies to sound, mechanical, and electromagnetic waves. The higher the frequency at a fixed velocity — the shorter the wavelength, and vice versa.

5. Resonance and Quality Factor

Quality Factor of a Resonant System:

Q = \frac{ω_{0}}{Δ}

What is resonance?

Resonance occurs when the frequency of an external influence matches the natural frequency of the system. The quality factor shows how "sharply" the system responds to this frequency: Q = ω₀ / Δω — the higher, the sharper the peak and narrower the resonance band.

Mechanical Pendulum:

f = \frac{1}{2 π} \cdot \sqrt{\frac{g}{L}}

Application

A swing is a classic example of a mechanical resonator. The frequency depends on the length of the string L, but not on the mass of the body.

Stretched String:

f = \frac{1}{2 L} \cdot \sqrt{\frac{T}{ρ}}

Application

The frequency of the fundamental harmonic depends on the length of the string L, tension T, and linear density ρ. This is the basis of how musical instruments produce sound.

Oscillatory Circuit (LC circuit):

f = \frac{1}{2 π \cdot \sqrt{L \cdot C}}

Application

Electrical circuits with inductance L and capacitance C are tuned to the desired frequency. This is the basis of filters, radio receivers, and generators.

6. Damped Oscillations

Differential Equation of Damped Oscillations:

\ddot{x} + 2 ζ ω_{0} \dot{x} + ω_{0}^{2} x = 0

System Parameters

ω₀ = √(k / m) — natural frequency without damping
ζ = c / (2√(km)) — damping coefficient

This equation describes the motion of any system with viscous friction: mechanical, acoustic, electrical. The form of the solution depends on the value of ζ.

Weak Damping (ζ < 1):

x (t) = A e^{- ζ ω_{0 > t}} \cos (ω_{d > t + φ)}

Physical Meaning

The amplitude of oscillations decreases exponentially with time. The oscillation frequency is slightly lower than that of an undamped system:

ω_d = ω₀ · √(1 – ζ²)

This mode is realized in most practical problems: sound in air, electrical circuits, oscillations with slight friction.

Formula for the Frequency of Damped Oscillations:

ω_{d > = ω_{0 > \sqrt{1 - ζ^{2}}}}

Physical Meaning

In damped oscillations, the frequency decreases compared to the natural frequency of the system. This is due to energy loss (e.g., due to friction or resistance).

If the damping coefficient ζ = 0, the frequency remains unchanged: ω_d = ω₀.

As ζ increases, the frequency decreases, and at ζ → 1, the system enters critical damping, where oscillations disappear.

The formula allows for a preliminary assessment of how damping affects system dynamics and for selecting optimal parameters in engineering tasks.

Logarithmic Decrement of Damping:

Λ = \ln (\frac{A_{n}}{A_{n + 1}}}) = \frac{2 π ζ}{\sqrt{1 - ζ^{2}}}

Physical Meaning

The logarithmic decrement shows how quickly the amplitude of a damped oscillation decreases: how much smaller the next peak becomes compared to the previous one.

The value Λ is related to the damping coefficient ζ:
Λ = (2π·ζ) / √(1 – ζ²)

This characteristic is important in the analysis of weakly damped systems, especially in mechanics, electronics, and acoustics.

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