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Wave Resonance Calculator

Analyze wave resonance phenomena, amplitude amplification, and damping effects

Parameters

mⓘ
Hzⓘ
mⓘ
m/sⓘ
ⓘ
Show Trail

Controls

xⓘ

Calculated Values

Fundamental Frequency:
85.00;Hz85.00;Hz
Harmonic Number:
0.00;0.00;
Resonance Factor:
0.50;0.50;
Effective Amplitude:
0.50;m0.50;m
Quality Factor:
10.00;10.00;

Examples

Example 1: Guitar String

Guitar string with length 0.65 m and wave speed 400 m/s.

  • Fundamental Frequency: 308.00308.00
  • Harmonic Number: 1.001.00
  • Resonance Factor: 1.001.00

Example 2: Organ Pipe

Organ pipe with length 2.0 m and wave speed 340 m/s.

  • Fundamental Frequency: 85.0085.00
  • Harmonic Number: 2.002.00
  • Resonance Factor: 0.500.50

Example 3: Microwave Cavity

Microwave cavity with length 0.1 m and wave speed 3×10⁸ m/s.

  • Fundamental Frequency: 1500000000.001500000000.00
  • Harmonic Number: 1.001.00
  • Resonance Factor: 1.001.00

Visualization

Wave Resonance

Resonance occurs when a system is driven at one of its natural frequencies, leading to maximum amplitude response. In wave systems, resonance happens when the driving frequency matches a natural frequency of the system.

For a string or air column with fixed ends, the natural frequencies are fₙ = nv/(2L), where n is the harmonic number, v is the wave speed, and L is the length. The fundamental frequency (n=1) is the lowest resonance frequency.

At resonance, the amplitude of oscillation increases dramatically due to constructive interference between the driving force and the system's natural oscillations. The maximum amplitude depends on the damping coefficient.

Damping reduces the amplitude over time and broadens the resonance peak. The quality factor Q = ω₀/γ describes the sharpness of the resonance, where ω₀ is the natural frequency and γ is the damping coefficient.

Resonance is fundamental to many physical phenomena, including musical instruments, radio tuning, structural vibrations, and quantum mechanical systems.

Key Concepts

  • Resonance Frequency: Natural frequency of the system
  • Amplitude Amplification: Increase in amplitude at resonance
  • Damping: Energy loss that reduces amplitude over time
  • Quality Factor: Measure of resonance sharpness
  • Harmonics: Integer multiples of fundamental frequency
  • Driving Force: External force that excites the system
  • Natural Frequency: Inherent frequency of the system
  • Constructive Interference: Amplification at resonance

Real-World Applications

  • Musical Instruments: String and wind instruments
  • Radio and TV: Tuning to specific frequencies
  • Structural Engineering: Avoiding resonance in buildings
  • Medical Imaging: Ultrasound and MRI
  • Quantum Physics: Atomic and molecular resonances

Explore Further

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  • Standing Waves

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  • Wave Reflection

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  • Wave Transmission

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Physics Equations

Fundamental Frequency:
f1=v2Lf_1 = \frac{v}{2L}
Harmonic Frequencies:
fn=nf1=nv2Lf_n = nf_1 = \frac{nv}{2L}
Resonance Factor:
R=11+∣f−f1∣f1R = \frac{1}{1 + \frac{|f - f_1|}{f_1}}
Quality Factor:
Q=ω0γQ = \frac{\omega_0}{\gamma}
Damped Amplitude:
A(t)=A0e−γtA(t) = A_0e^{-\gamma t}

Step-by-Step Solution

See how the main results are calculated.

1

Step 1: Calculate Fundamental Frequency

First, we calculate the fundamental frequency of the system:

Equation:

f1=v2Lf_1 = \frac{v}{2L}

Calculation:

f1=3402×2.00=85.00 Hzf_1 = \frac{340}{2 \times 2.00} = 85.00 \text{ Hz}

Explanation:

The fundamental frequency is the lowest natural frequency of the system.

2

Step 2: Determine Harmonic Number

Calculate which harmonic corresponds to the given frequency:

Equation:

n=ff1n = \frac{f}{f_1}

Calculation:

n=1.0085.00=0n = \frac{1.00}{85.00} = 0

Explanation:

The harmonic number indicates which mode of vibration is excited.

3

Step 3: Calculate Resonance Factor

The resonance factor determines amplitude amplification:

Equation:

R=11+∣f−f1∣f1R = \frac{1}{1 + \frac{|f - f_1|}{f_1}}

Calculation:

R=11+∣1.00−85.00∣85.00=0.503R = \frac{1}{1 + \frac{|1.00 - 85.00|}{85.00}} = 0.503

Explanation:

The resonance factor is maximum (1) when the driving frequency equals the natural frequency.

4

Step 4: Calculate Effective Amplitude

The effective amplitude is the product of base amplitude and resonance factor:

Equation:

Aeff=A0×RA_{eff} = A_0 \times R

Calculation:

Aeff=1.000×0.503=0.5030 mA_{eff} = 1.000 \times 0.503 = 0.5030 \text{ m}

Explanation:

The effective amplitude shows how much the resonance amplifies the base amplitude.

Frequently Asked Questions (FAQ)

What is resonance?

Resonance occurs when a system is driven at one of its natural frequencies, leading to maximum amplitude response. It happens when the driving frequency matches a natural frequency of the system.

How does damping affect resonance?

Damping reduces the amplitude over time and broadens the resonance peak. Higher damping means lower maximum amplitude and a less sharp resonance peak.

What is the quality factor?

The quality factor (Q) describes the sharpness of the resonance peak. Higher Q means a sharper, more selective resonance. Q = ω₀/γ, where ω₀ is the natural frequency and γ is the damping coefficient.

What are harmonics?

Harmonics are integer multiples of the fundamental frequency. For a string with fixed ends, the harmonics are fₙ = nf₁, where n = 1, 2, 3... The fundamental frequency (n=1) is the lowest resonance frequency.

Why does amplitude increase at resonance?

At resonance, the driving force is in phase with the system's natural oscillations, leading to constructive interference and maximum energy transfer. This results in dramatic amplitude amplification.

Practice MCQs

  1. The fundamental frequency of a string with fixed ends is:
  2. At resonance, the amplitude:
  3. Higher damping results in:
  4. The quality factor is:
  5. Harmonics are:

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