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Linking Speed, Frequency, and Wavelength · Waves, Fields, and Electromagnetism · HS-PS4-1 · wave equation

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Linking Speed, Frequency, and Wavelength

with Atlas

Atlas stands on a long pier over calm ocean water, holding a stopwatch and a measuring tape, watching a series of evenly spaced waves roll steadily beneath the dock — counting crests as they pass and pacing out the distance between them.

Explain what frequency, wavelength, and wave speed each measure and how their units relate.
Apply the wave equation v = fλ to calculate any one quantity when the other two are given.
Predict how wavelength changes when frequency increases while wave speed stays constant.
Calculate wave speed, frequency, or wavelength in different media using v = fλ when two of the three quantities are given.
Identify and reject the misconception that a higher-frequency wave travels faster through the same medium.

Key terms

Wave speed: How fast a wave pattern advances through a medium, measured in meters per second.
Frequency: The number of complete wave cycles passing a fixed point each second, measured in hertz.
Wavelength: The distance between two consecutive identical points on a wave, such as crest to crest.
Medium: The material or space through which a wave travels, whose properties set the wave speed.
Period: The time for one complete wave cycle, equal to the reciprocal of the frequency.

Reading the Wave Equation

The wave equation v = fλ ties together three measurable quantities. Frequency counts cycles per second, wavelength measures the spatial length of one cycle, and their product gives the distance the pattern travels per second, which is the speed. Because the equation has three variables, knowing any two determines the third. Checking units confirms the logic: hertz is cycles per second and meters is meters per cycle, so multiplying cancels cycles and leaves meters per second, the unit of speed.

The Medium Sets the Speed

A subtle but vital point is that wave speed in a given medium is fixed by the properties of that medium — the tension and linear density of a string, or the elasticity and density of air — and not by the source. Consequently, if you change the frequency, the wavelength must change inversely to keep v constant: double the frequency and the wavelength halves. This is why a soprano and a bass note travel through the same room air at the same roughly 340 m/s, differing only in wavelength, and why a higher pitch never reaches your ears faster.

Worked examples

A sound wave travels at 340 m/s with a frequency of 680 Hz. Find its wavelength.

Rearrange v = fλ to solve for wavelength: λ = v / f.
Substitute the known values: λ = 340 m/s / 680 Hz.
Divide to obtain λ = 0.50.
The units m/s divided by 1/s give meters.

Answer: 0.50 m.

Light travels through glass at 2.0 × 10⁸ m/s with frequency 5.0 × 10¹⁴ Hz. Find its wavelength in the glass.

Use λ = v / f with the in-glass speed.
Substitute: λ = (2.0 × 10⁸ m/s) / (5.0 × 10¹⁴ Hz).
Divide coefficients (2.0/5.0 = 0.40) and subtract exponents (8 − 14 = −6).
Obtain λ = 0.40 × 10⁻⁶ m = 4.0 × 10⁻⁷ m (400 nm).

Answer: 4.0 × 10⁻⁷ m (400 nm inside the glass).

Every wave — whether it is a ripple on a pond, a sound traveling through air, or a radio signal crossing a city — obeys one elegant relationship: v = fλ. Here, v is the wave speed (meters per second), f is the frequency (hertz, or cycles per second), and λ (lambda) is the wavelength (meters, the distance between two consecutive crests or two consecutive compressions). Think of it this way. Frequency tells you how many complete wave cycles pass a fixed point every second. Wavelength tells you how long each cycle is in space. Multiply them and you get the total distance the wave pattern advances each second — which is exactly the speed. The constraint this imposes is powerful: for a wave traveling through a given medium, the speed is fixed by the properties of that medium (the tension and density of a string, the elasticity and density of air, and so on). That means if you change the frequency, the wavelength must change in the opposite direction to keep v constant. Double the frequency, and the wavelength is cut in half. Halve the frequency, and the wavelength doubles. Speed, frequency, and wavelength are locked in a three-way balance. A common trap: students assume that a higher-frequency wave must be moving faster. That is not true in the same medium — the speed does not change. What changes is how 'tightly packed' the wave pattern is. A soprano's voice and a bass guitar note both travel through the same room air at roughly 340 m/s; the soprano's wave simply has a much shorter wavelength. Keep that distinction sharp and v = fλ becomes one of the most versatile tools in physics.

Activity

Drag each wave scenario card to the correct calculated value on the answer board.

Practice

A water wave has a wavelength of 3.0 m and travels at 12 m/s; calculate its frequency using the wave equation.

Explain why doubling the frequency of a wave in the same medium halves its wavelength but leaves the speed unchanged.

Common mistakes to avoid

Higher-frequency waves travel faster in the same medium.In a given medium the speed is fixed by the medium, so raising frequency only shortens the wavelength, never increases the speed.
Changing the source loudness or amplitude changes the wave speed.Amplitude affects energy carried, not speed; wave speed depends only on the properties of the medium.

Check your understanding

A sound wave travels through air at 340 m/s with a frequency of 680 Hz. What is its wavelength?

A guitar string produces a note at 440 Hz. A second string in the same room produces a note at 880 Hz. Compared with the 440 Hz wave, the 880 Hz wave travels through the air at

Light travels through glass at 2.0 × 10⁸ m/s. If the frequency of the light is 5.0 × 10¹⁴ Hz, what is its wavelength inside the glass?

Recap

The wave equation v = fλ links speed, frequency, and wavelength so any two determine the third. In a given medium the speed is fixed, so frequency and wavelength vary inversely, and a higher-frequency wave has a shorter wavelength but travels at the same speed.

Reflect

Where do you notice the same wave speed carrying very different pitches or colors?