2D Wave equation

$$y=A\cos \left(\frac{2\pi }{\lambda }x\pm \frac{2\pi }{T}t+\varphi \right)$$

Wave Equation Formulation

1. Function in Space

The amplitude is maximum at $(x=0)$, implying the function is cosine.

Initial form: $y=A\cos \theta$

The cosine resets at intervals of $2\pi$, i.e., $x=\lambda$ (wavelength): $y=A\cos \left(\frac{2\pi }{\lambda }x\right)$

2. Function in Time

The cosine resets at intervals of $2\pi$, i.e., $t=T$ (time period): $y=A\cos \left(\frac{2\pi }{\lambda }x\pm \frac{2\pi }{T}t\right)$

Note: The temporal part gets a $\pm$ because the wave could be moving forward or backward.

3. Final Equation

$\boxed{y=A\cos \left(\frac{2\pi }{\lambda }x\pm \frac{2\pi }{T}t+\varphi \right)}$

Where:

• $A$: Amplitude
• $\lambda$: Wavelength
• $T$: Time period
• $\varphi$: Phase shift
• $x$: Position in space
• $t$: Time

---

References

The equation of a wave | Physics | Khan Academy