Check the dimensional correctness of y = A sin(ωt - kx)
Travelling wave. A wave moving along x. The angle inside the sine must be a pure number, and it is.
Equation checker
Matching dimensions are necessary, not sufficient. A check cannot see pure numbers such as 1/2 or 2π, so a consistent equation can still be wrong.
y = A sin(ω t - k x)y[L]=A sin(omega t - k x)[L]Step by step
1. What each symbol stands for
- y displacement [L]
- A amplitude [L]
- ω angular frequency [T-1]
- t time [T]
- k wave number [L-1]
- x displacement [L]
2. The dimensions of each term
- y [L] [L]
- A sin(omega t - k x) [L] × sin([T-1] × [T] - [L-1] × [L]) [L]
3. Compare
Every term has dimensions [L], so the equation is dimensionally consistent.
What it proves and what it does not
It can show
- Every term that is added or subtracted has the same dimensions, so the sum makes sense.
- Both sides of the equals sign have the same dimensions.
- Angles inside sin, cos and similar functions are pure numbers.
It cannot show
- A pure number such as 1/2 or 2π has no dimensions, so a check cannot see a missing or wrong one.
- Two quantities with the same dimensions, such as work and torque, cannot be told apart.
- A sign error or a wrong power that keeps the dimensions equal goes unnoticed.
Matching dimensions are necessary but not sufficient. Pure numbers such as 1/2 or 2π have no dimensions, so a check cannot see them, and an equation with matching dimensions can still be wrong.
Look up each quantity
y = A sin(ωt - kx): quick answers
Is y = A sin(ωt - kx) dimensionally correct?
Yes. Every term has the same dimensions, [L], so y = A sin(ωt - kx) is dimensionally consistent. That does not prove the equation itself is correct.
Does a dimension check prove y = A sin(ωt - kx) is right?
No. A check can show that an equation is wrong, but it cannot show that it is right. Pure numbers such as 1/2 and 2π have no dimensions, so the check cannot see them.