Identify wave number from equation:
Comparing \(y = 2\cos 2\pi(10t - 0.0080x + 0.35)\) with standard form \(y = A\cos\!\left(\omega t - kx + \phi_0\right)\):
We get \(k = 2\pi \times 0.0080\) cm⁻¹ (since the equation has \(2\pi \times 0.0080x\))
Wavelength: \(\lambda = \dfrac{1}{0.0080} = 125\) cm
Convert separation to cm:
\(\Delta x = 0.5\) m \(= 50\) cm
Calculate phase difference:
$$\Delta\phi = \frac{2\pi}{\lambda}\times\Delta x = \frac{2\pi}{125}\times 50 = \frac{100\pi}{125} = 0.8\pi \text{ rad}$$
Answer: 0.8π rad ✓
The general equation of a sinusoidal travelling wave moving in the +x direction is:
\(y(x,t) = A\sin(\omega t - kx + \phi_0)\)
or \(y(x,t) = A\cos(\omega t - kx + \phi_0)\)
Where A = amplitude, ω = angular frequency = 2πf, k = wave number = 2π/λ, φ₀ = initial phase. For a wave moving in −x direction, the sign of kx is positive: y = Asin(ωt + kx).
📌 From y = 2cos2π(10t − 0.0080x + 0.35):
📌 Amplitude A = 2.0 cm
📌 Frequency f = 10 Hz (coefficient of t)
📌 ω = 2π × 10 = 20π rad/s
📌 1/λ = 0.0080 cm⁻¹ → λ = 125 cm = 1.25 m
📌 k = 2π × 0.0080 = 0.016π rad/cm
📌 Wave speed v = fλ = 10 × 125 = 1250 cm/s = 12.5 m/s
📌 Initial phase φ₀ = 2π × 0.35 = 0.7π rad
For two points at the same time t, separated by distance Δx:
\(\Delta\phi = k\cdot\Delta x = \dfrac{2\pi}{\lambda}\cdot\Delta x\)
For the same point at two different times separated by Δt:
\(\Delta\phi = \omega\cdot\Delta t = 2\pi f\cdot\Delta t\)
Phase difference = 2π means the points are one full wavelength apart (or one full time period apart) — they are in phase. Phase difference = π means they are completely out of phase.
In this problem, x is in cm but the separation is given in metres (0.5 m = 50 cm). Always convert to the same unit before calculating. This unit mismatch is a very common NEET trap — a student who uses Δx = 0.5 directly gets 0.008π which is wrong. Always check units!
v = fλ = ω/k
All three of v, f, λ are related by this equation. In this problem: v = fλ = 10 × 1.25 = 12.5 m/s. Also v = ω/k = 20π/(0.016π) = 1250 cm/s = 12.5 m/s ✓. The wave speed depends on the medium, not on frequency or amplitude.