  Here we report the measurements of two-dimensional (2-D) spectra of the
streamwise velocity ($u$) in a high Reynolds number turbulent boundary layer. A
novel experiment employing multiple hot-wire probes was carried out at friction
Reynolds numbers ranging from 2400 to 26000. Taylor's frozen turbulence
hypothesis is used to convert temporal-spanwise information into a 2-D spatial
spectrum which shows the contribution of streamwise ($\lambda_x$) and spanwise
($\lambda_y$) length scales to the streamwise variance at a given wall height
($z$). At low Reynolds numbers, the shape of the 2-D spectra at a constant
energy level shows $\lambda_y/z \sim (\lambda_x/z)^{1/2}$ behaviour at larger
scales, which is in agreement with the existing literature at a matched
Reynolds number obtained from direct numerical simulations. However, at high
Reynolds numbers, it is observed that the square-root relationship tends
towards a linear relationship ($\lambda_y \sim \lambda_x$) as required for
self-similarity and predicted by the attached eddy hypothesis.
