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Wave Reflection Calculator

Analyze wave reflection at boundaries, reflection coefficients, and phase shifts

Parameters

mⓘ
Hzⓘ
mⓘ
kg/m²sⓘ
kg/m²sⓘ
Show Trail

Controls

xⓘ

Calculated Values

Reflection Coefficient:
0.33;0.33;
Transmission Coefficient:
0.67;0.67;
Reflected Amplitude:
0.33;m0.33;m
Transmitted Amplitude:
0.67;m0.67;m
Phase Shift:
0.00;°0.00;°

Examples

Example 1: Sound Wave Reflection

Sound wave reflecting from air (Z=415) to water (Z=1.48×10⁶ kg/m²s).

  • Reflection Coefficient: 1.001.00
  • Transmission Coefficient: 0.000.00
  • Phase Shift: 0.000.00

Example 2: String Fixed End

Wave on string reflecting from fixed end (Z₂ → ∞).

  • Reflection Coefficient: 0.920.92
  • Transmission Coefficient: 0.080.08
  • Phase Shift: 180.00180.00

Example 3: Impedance Matching

Wave with matched impedances (Z₁ = Z₂).

  • Reflection Coefficient: 0.000.00
  • Transmission Coefficient: 1.001.00
  • Phase Shift: 0.000.00

Visualization

Wave Reflection

Wave reflection occurs when a wave encounters a boundary between two different media or when it reaches a fixed or free end. The behavior of the reflected wave depends on the properties of both media and the type of boundary.

The reflection coefficient (R) determines the amplitude of the reflected wave relative to the incident wave. It is given by R = (Z₂ - Z₁)/(Z₂ + Z₁), where Z₁ and Z₂ are the impedances of the two media.

When a wave reflects from a fixed end (high impedance), it undergoes a phase shift of π radians (180°), meaning the reflected wave is inverted. When reflecting from a free end (low impedance), there is no phase shift.

The transmission coefficient (T) determines the amplitude of the transmitted wave: T = 2Z₁/(Z₁ + Z₂). The sum of the reflection and transmission coefficients squared equals 1, ensuring energy conservation.

Wave reflection is fundamental to understanding phenomena such as echoes, standing waves, impedance matching in electronics, and the behavior of light at optical interfaces.

Key Concepts

  • Reflection Coefficient: Ratio of reflected to incident wave amplitude
  • Transmission Coefficient: Ratio of transmitted to incident wave amplitude
  • Impedance: Product of medium density and wave speed
  • Phase Shift: Change in phase angle upon reflection
  • Fixed End: High impedance boundary causing π phase shift
  • Free End: Low impedance boundary with no phase shift
  • Energy Conservation: R² + T² = 1
  • Boundary Conditions: Constraints at media interfaces

Real-World Applications

  • Acoustics: Echo formation and room acoustics
  • Optics: Light reflection at mirrors and interfaces
  • Electronics: Signal reflection in transmission lines
  • Seismology: Earthquake wave reflection at boundaries
  • Engineering: Structural vibration analysis

Explore Further

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  • Standing Waves

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  • Wave Transmission

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  • Wave Diffraction

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Physics Equations

Reflection Coefficient:
R=Z2−Z1Z2+Z1R = \frac{Z_2 - Z_1}{Z_2 + Z_1}
Transmission Coefficient:
T=2Z1Z1+Z2T = \frac{2Z_1}{Z_1 + Z_2}
Energy Conservation:
R2+T2=1R^2 + T^2 = 1
Reflected Amplitude:
Ar=RAiA_r = RA_i
Transmitted Amplitude:
At=TAiA_t = TA_i

Step-by-Step Solution

See how the main results are calculated.

1

Step 1: Calculate Reflection Coefficient

First, we calculate the reflection coefficient using the impedance values:

Equation:

R=Z2−Z1Z2+Z1R = \frac{Z_2 - Z_1}{Z_2 + Z_1}

Calculation:

R=800−400800+400=0.333R = \frac{800 - 400}{800 + 400} = 0.333

Explanation:

The reflection coefficient determines the amplitude and phase of the reflected wave.

2

Step 2: Calculate Transmission Coefficient

Next, we calculate the transmission coefficient:

Equation:

T=2Z1Z1+Z2T = \frac{2Z_1}{Z_1 + Z_2}

Calculation:

T=2×400400+800=0.667T = \frac{2 \times 400}{400 + 800} = 0.667

Explanation:

The transmission coefficient determines the amplitude of the transmitted wave.

3

Step 3: Calculate Reflected Amplitude

The amplitude of the reflected wave is:

Equation:

Ar=RAiA_r = RA_i

Calculation:

Ar=0.333×1.000=0.3333 mA_r = 0.333 \times 1.000 = 0.3333 \text{ m}

Explanation:

The reflected wave amplitude is the product of reflection coefficient and incident amplitude.

4

Step 4: Determine Phase Shift

The phase shift depends on the sign of the reflection coefficient:

Equation:

ϕ={0°if R>0180°if R<0\phi = \begin{cases} 0° & \text{if } R > 0 \\ 180° & \text{if } R < 0 \end{cases}

Calculation:

ϕ=0°\phi = 0°

Explanation:

Negative reflection coefficient indicates a 180° phase shift (wave inversion).

Frequently Asked Questions (FAQ)

What is the reflection coefficient?

The reflection coefficient (R) is the ratio of the amplitude of the reflected wave to the amplitude of the incident wave. It ranges from -1 to +1, where negative values indicate a phase shift of 180°.

When does a wave undergo a phase shift upon reflection?

A wave undergoes a phase shift of 180° when reflecting from a boundary with higher impedance (fixed end) or when the reflection coefficient is negative. No phase shift occurs when reflecting from a lower impedance boundary (free end).

What is impedance matching?

Impedance matching occurs when the impedances of two media are equal (Z₁ = Z₂). In this case, the reflection coefficient is zero, meaning no reflection occurs and all energy is transmitted.

How does energy conservation apply to wave reflection?

Energy conservation requires that R² + T² = 1, where R is the reflection coefficient and T is the transmission coefficient. This ensures that the total energy of the incident wave equals the sum of reflected and transmitted wave energies.

What is the difference between fixed and free end reflection?

At a fixed end (high impedance), the wave reflects with a phase shift of 180° and the displacement is always zero. At a free end (low impedance), there is no phase shift and the wave reflects with the same phase.

Practice MCQs

  1. The reflection coefficient for a wave at a fixed end is:
  2. When Z₁ = Z₂, the reflection coefficient is:
  3. The energy conservation equation for wave reflection is:
  4. A wave reflecting from a free end has:
  5. The transmission coefficient is always:

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