For a travelling harmonic wave y(x,t) = 2.0cos2π(10t°0.0080x +0.35), where x and y are in cm and t in s. The phase difference between oscillatory motion of two points separated by a distance of 0.5 m is: ____.

We are given the travelling harmonic wave

$y(x,t)=2.0\backslash cos\backslash !\backslash bigl[2\backslash pi\backslash ,(10t\; -\; 0.0080x\; +\; 0.35)\backslash bigr]$

where x and y are measured in centimetres (cm) and t in seconds (s).

The term that contains x determines how the phase varies with position. Comparing the argument of the cosine with the standard form

$\backslash cos\backslash !\backslash bigl[2\backslash pi\backslash ,(ft\; -\; \backslash frac\{k\}\{2\backslash pi\}x\; +\; \backslash phi\_0)\backslash bigr]$

we identify the spatial coefficient as

$\backslash frac\{k\}\{2\backslash pi\}=0.0080\backslash ;\backslash text\{cm\}^\{-1\}$

Hence the wave‑number k is

$k\; =\; 2\backslash pi\; \backslash times\; 0.0080\; =\; 0.016\backslash pi\backslash ;\backslash text\{rad cm\}^\{-1\}$

The phase difference Δφ between two points separated by a distance Δx is given by

$\backslash Delta\backslash phi\; =\; k\backslash ,\backslash Delta\; x$

Convert the separation from metres to centimetres: 0.5 m = 50 cm.

Substituting k and Δx**:

$\backslash Delta\backslash phi\; =\; (0.016\backslash pi\backslash ;\backslash text\{rad cm\}^\{-1\})\backslash times\; 50\backslash ;\backslash text\{cm\}$

Calculate the product:

$\backslash Delta\backslash phi\; =\; 0.016\backslash pi\; \backslash times\; 50\; =\; 0.8\backslash pi\backslash ;\backslash text\{rad\}$

Therefore the phase difference between the oscillatory motion of the two points is

$0.8\backslash pi\backslash ;\backslash text\{rad\}$