Universal Wave Equation

Physics Notes — Wave Equation and Wave Velocity

1. Wave Properties Review

Wavelength (λ\lambdaλ)

Definition:

Distance between two identical points on consecutive waves.

Examples:

  • crest to crest

  • trough to trough

Measured in:

  • meters (m)

Frequency (fff)

Definition:

Number of wave cycles per second.

Measured in:

  • hertz (Hz)

1 Hz=1 cycle/second1\text{ Hz}=1\text{ cycle/second}1 Hz=1 cycle/second

Period (TTT)

Definition:

Time required for one complete wave cycle.

Measured in:

  • seconds (s)

Frequency and Period Relationship

Inverse relationship:

T=1fT=\frac{1}{f}T=f1​

TTT

s

T es el período; f es la frecuencia.

f=1T=12=0.5 Hzf=\frac{1}{T}=\frac{1}{2}=0.5\,\text{Hz}f=T1​=21​=0.5Hz

Tf = 0.5 Hz

f=1Tf=\frac{1}{T}f=T1​

TTT

s

T es el período; f es la frecuencia.

f=1T=12=0.5 Hzf=\frac{1}{T}=\frac{1}{2}=0.5\,\text{Hz}f=T1​=21​=0.5Hz

Tf = 0.5 Hz

2. Wave Speed

Wave Velocity (vvv)

Definition:

Speed at which a wave travels.

General speed formula:

v=dtv=\frac{d}{t}v=td​

For waves:

  • distance = wavelength (λ\lambdaλ)

  • time = period (TTT)

So:

v=λTv=\frac{\lambda}{T}v=Tλ​

Since:

T=1fT=\frac1fT=f1​

we get the universal wave equation:

v=λfv=\lambda fv=λf

3. Universal Wave Equation

Main Formula

v=λfv=\lambda fv=λf

Where:

  • vvv = wave speed (m/s)

  • λ\lambdaλ = wavelength (m)

  • fff = frequency (Hz)

Rearranged Formulas

Solve for Wavelength

λ=vf\lambda=\frac{v}{f}λ=fv​

Solve for Frequency

f=vλf=\frac{v}{\lambda}f=λv​

4. Example Problems

Example 1 — Solve for Velocity

Given:

  • f=440 Hzf=440\text{ Hz}f=440 Hz

  • λ=0.79 m\lambda=0.79\text{ m}λ=0.79 m

Use:

v=λfv=\lambda fv=λf

v=0.79(440)v=0.79(440)v=0.79(440) v=347.6 m/sv=347.6\text{ m/s}v=347.6 m/s

Example 2 — Solve for Wavelength

Given:

  • v=3.0×108 m/sv=3.0\times10^8\text{ m/s}v=3.0×108 m/s

  • f=1.08×108 Hzf=1.08\times10^8\text{ Hz}f=1.08×108 Hz

Use:

λ=vf\lambda=\frac{v}{f}λ=fv​

λ=3.0×1081.08×108\lambda=\frac{3.0\times10^8}{1.08\times10^8}λ=1.08×1083.0×108​ λ=2.8 m\lambda=2.8\text{ m}λ=2.8 m

Example 3 — Solve for Frequency

Given:

  • v=340 m/sv=340\text{ m/s}v=340 m/s

  • λ=0.66 m\lambda=0.66\text{ m}λ=0.66 m

Use:

f=vλf=\frac{v}{\lambda}f=λv​

f=3400.66f=\frac{340}{0.66}f=0.66340​ f=515 Hzf=515\text{ Hz}f=515 Hz

5. Relationships Between Wave Properties

Velocity and Wavelength

Direct relationship.

  • speed increases → wavelength increases

  • speed decreases → wavelength decreases

Frequency and Wavelength

Inverse relationship.

  • frequency increases → wavelength decreases

  • frequency decreases → wavelength increases

6. Wave Velocity Depends on Medium

Important Rule

Wave speed depends ONLY on properties of the medium.

NOT on:

  • amplitude

  • frequency

  • wavelength

Medium Properties Affecting Velocity

  • density

  • tension

  • temperature

  • stiffness

  • compressibility

  • pressure

7. Tension and Wave Speed

Tension

Definition:

Stretching force in strings/springs.

Effect

Higher tension:

  • faster wave speed

Lower tension:

  • slower wave speed

Applies To

  • strings

  • springs

  • cables

8. Density and Wave Speed

Density

Definition:

Mass per unit volume or length.

Effect in Springs and Gases

Higher density:

  • slower wave speed

Lower density:

  • faster wave speed

Reason:
Dense particles resist motion more (greater inertia).

Important Exception

Density alone cannot predict wave speed in all substances.

Example:

  • water denser than air

  • sound still travels faster in water

9. Temperature and Wave Speed

In Gases

Higher temperature:

  • particles move faster

  • collisions transfer energy faster

  • wave speed increases

Example

Sound travels faster in:

  • warm air
    than in:

  • cold air

10. Entering a New Medium

When a wave enters another medium:

Frequency

Stays constant.

Velocity

Changes.

Wavelength

Changes.

Why?

Using:

v=λfv=\lambda fv=λf

If frequency stays constant and velocity changes:

  • wavelength must also change

Boundary Effects

Part of wave:

  • transmitted forward

Part of wave:

  • reflected backward

Ocean Wave Example

Deep water:

  • higher speed

  • longer wavelength

Shallow water:

  • lower speed

  • shorter wavelength

Sound Example

Sound moving from:

  • cold air
    to

  • warm air

Results:

  • wave speed increases

  • wavelength increases

11. Important Vocabulary

Term

Meaning

Wave speed

Speed of wave movement

Wavelength

Length of one cycle

Frequency

Cycles per second

Period

Time for one cycle

Tension

Stretching force

Density

Mass per volume/length

Boundary

Separation between media

Reflection

Wave bouncing back

Transmission

Wave continuing forward

Most Important Equations

Universal Wave Equation

v=λfv=\lambda fv=λf

Frequency and Period

T=1fT=\frac{1}{f}T=f1​

TTT

s

T es el período; f es la frecuencia.

f=1T=12=0.5 Hzf=\frac{1}{T}=\frac{1}{2}=0.5\,\text{Hz}f=T1​=21​=0.5Hz

Tf = 0.5 Hz

f=1Tf=\frac{1}{T}f=T1​

TTT

s

T es el período; f es la frecuencia.

f=1T=12=0.5 Hzf=\frac{1}{T}=\frac{1}{2}=0.5\,\text{Hz}f=T1​=21​=0.5Hz

Tf = 0.5 Hz

Most Important Concepts

Frequency Stays Constant Across Media

Velocity and wavelength change.

Wave Speed Depends on Medium

Not wave properties.

High Tension

→ faster waves

High Density

→ slower waves (in springs/gases)

High Temperature

→ faster sound in gases

Most Likely Test Questions

  • Use v=λfv=\lambda fv=λf

  • Solve for frequency/wavelength/speed

  • Explain medium effects on wave speed

  • Explain what changes when wave enters new medium

  • Relationship between wavelength and frequency

  • Relationship between wavelength and velocity

  • Why ocean waves shorten near shore

  • Why sound moves faster in warm air