Why is a rightward-traveling wave expressed as $f(x,t) = f$(x - vt, 0)?

Conditions

• The wave propagates in the positive x-direction.
• The propagation speed is constant $v$.
• The waveform remains unchanged during propagation.

Reasoning, step by step

1. Identify the initial wave profile at $t=0$ as $f(x,0)$.
2. Calculate the distance the wave travels to the right after time $t$, which is $vt$.
3. Apply the rule for horizontal translation of function graphs: shifting a graph to the right by $vt$ corresponds to replacing the independent variable $x$ with $x-vt$.
4. Equate the wave profile at time $t$, $f(x,t)$, with the shifted initial profile $f(x-vt,0)$.

Example

The video compares the waveform at $t=0$ with the waveform after time $t$. The subtitles state: "After time t, the wave has moved a distance vt to the right" and "The wave's shape can also be expressed as f(x - vt, 0)."

Common misconceptions

• Mistakenly thinking that the minus sign in $x-vt$ indicates the wave is propagating to the left. The minus sign arises from the function translation rule for shifting the graph to the right.
• Confusing the wave's propagation speed $v$ with the velocity of the particles on the rope, which only move vertically.

Watch the explanation

Explore next

• How does the video demonstrate that a wave maintains its shape while propagating?
• Why can we not describe a wave using only $y = f(x)$, but must instead use $y = f(x, t)$?