How do you check that the argument of the sine function in the wave equation has consistent units?

Conditions

• Applies to the traveling sinusoidal wave equation $y(x,t)=A\sin(kx-\omega t+\phi)$.
• $k$ has units rad/m.
• $x$ has units m.
• $\omega$ has units rad/s.
• $t$ has units s.
• $\phi$ has units rad.

Reasoning, step by step

1. Identify the argument of the sine function: $kx-\omega t+\phi$.
2. Assign the unit rad/m to $k$ and m to $x$.
3. Assign the unit rad/s to $\omega$ and s to $t$.
4. Assign the unit rad to $\phi$.
5. Multiply the units of $k$ and $x$: (rad/m) * m = rad.
6. Multiply the units of $\omega$ and $t$: (rad/s) * s = rad.
7. Conclude that each term in the argument carries units of radians, making the sine argument dimensionally consistent.

Example

The video displays a slide titled "Units" with the equation $y(x,t)=A\sin(kx-\omega t+\phi)$. A purple box highlights the argument $kx-\omega t+\phi$, and unit labels rad/m, m, rad/s, s, and rad are written beneath the corresponding factors. The speaker explains that the meters and seconds cancel out, leaving radians minus radians plus radians, or just radians overall.

Common misconceptions

• Assuming the sine function can accept arguments in degrees without conversion.
• Forgetting that the phase constant $\phi$ must also be in radians.
• Thinking that the units of $k$ and $\omega$ are dimensionless.

Watch the explanation

Explore next

• Why does the vertical position $y$ have the same units as the amplitude $A$?
• What is the standard equation for a traveling sinusoidal wave?