Why does the vertical position $y$ have the same units as the amplitude $A$?

Conditions

• Applies to the traveling sinusoidal wave equation $y(x,t)=A\sin(kx-\omega t+\phi)$.
• The sine function outputs a dimensionless number.
• $A$ is measured in distance units (e.g., meters).

Reasoning, step by step

1. Identify the wave equation $y(x,t)=A\sin(kx-\omega t+\phi)$.
2. Recognize that the sine function is dimensionless and unitless.
3. Conclude that the term $\sin(kx-\omega t+\phi)$ has no physical units.
4. Apply the principle of dimensional homogeneity: the units of the left side ($y$) must equal the units of the right side ($A$ times a dimensionless number).
5. Deduce that $y$ must have the same units as $A$.

Example

The video states, "But once it's done, the sine of that number in radians, there's no units on it anymore. Sine is a dimensionless, unitless type of quantity." It then concludes, "So that means whatever unit I have for the amplitude, like meters, I'm going to have the same unit for the vertical position on my wave, y."

Common misconceptions

• Thinking that the sine function carries the unit of radians into the final output.
• Assuming $y$ is dimensionless because it is a position on a graph.
• Believing that $A$ and $y$ can have different units as long as the equation balances numerically.

Watch the explanation

Explore next

• How do you check that the argument of the sine function in the wave equation has consistent units?
• What does the phase constant $\phi$ represent physically in the wave equation?