Physics · Waves · NEET
k measures how fast the phase of a wave changes as you move along the direction of travel. In the wave equation y = a sin(kx - wt), the term kx is the phase due to position. If you move a distance x = one wavelength (lambda), the phase must increase by one full cycle = 2 pi radians. So k x lambda = 2 pi, which gives k = 2 pi / lambda. In short, k = radians of phase gained per metre of path.
1/lambda is the ordinary wave number (called nu-bar), which counts how many complete waves fit in one metre. Angular wave number k multiplies that by 2 pi because it is measured in radians, and one complete wave = 2 pi radians. So k = 2 pi x (1/lambda) = 2 pi / lambda. Use k = 2 pi/lambda whenever the wave is written with sin or cos, because those functions work in radians.
The SI unit of angular wave number is radian per metre (rad/m), often just written per metre (1/m or m^-1). This matches NCERT. Since radian is dimensionless, kx must be a pure number (an angle in radians) with no unit, which is why k carries units of 1/length.
Write the equation as y = a sin(kx - wt) or y = a cos(kx - wt). The number multiplying x (the position variable) is k. For example in y = 0.005 sin(80x - 3t), k = 80 rad/m. Warning: check the unit of x first. If x is in cm and the equation is y = 2 cos[2 pi(10t - 0.0080x)], then k = 2 pi x 0.0080 per cm.
Phase difference between two points on a wave = k x (path difference), i.e. delta-phi = k x delta-x = (2 pi / lambda) x delta-x. This one formula is the most tested use of k in NEET. Two points one wavelength apart have delta-x = lambda, giving delta-phi = 2 pi (they are in phase).
For a travelling harmonic wave y(x, t) = 2.0 cos[2 pi(10t - 0.0080x + 0.35)], where x and y are in cm and t in s, the phase difference between the oscillatory motion of two points separated by 0.5 m is:
Try the real previous-year questions from this chapter — each with the answer and a full solution.
At NEET level treat k as a scalar magnitude given by 2 pi/lambda. (In higher physics k becomes a vector pointing in the direction the wave travels, but that is not needed for NEET numericals.)
Wave speed v = omega/k. Since omega = 2 pi f and k = 2 pi/lambda, this reduces to v = f x lambda. So k links directly to speed once you know omega.
Yes. Frequency f (and omega) stay the same across media, but speed and wavelength change, so k = 2 pi/lambda changes. If a wave speeds up, lambda increases and k decreases.
The magnitude k = 2 pi/lambda is always positive. A wave moving in the -x direction is written y = a sin(kx + wt); here k itself is still positive, only the sign inside the bracket differs.
