This question appears somewhat similar to other questions asking about why wavelength affects diffraction (a concept which I'm still not 100% sure on...) however my query is different and not answered that I can find. (The focus of my question is to what degree slit size affects diffraction in terms of the wavelength, not how or why) I was wondering, to what degree do the wavelength and the size of the slit have to be similar for diffraction to be reasonably observable (for example in the classic wave tank example) Does diffraction become negligible at 100x the wavelength? 1000x it? And is this different for longitudinal and transverse waves?
Answer
Whether the amount of diffraction is 'negligible' depends on how you define this criterion.
The first order minimum in the diffraction pattern from a single slit occurs where $\sin\theta = \lambda/d$ where $d$ is slit width, $\theta$ is diffraction angle and $\lambda$ is wavelength. If $d = \lambda$ the central lobe of the diffraction pattern will spread out $90$ degrees above and below the axis, filling the whole screen. If $d = 2\lambda$ the central lobe will spread to $30$ degrees above and below the axis. To achieve $\theta = 1\ \mathrm{degree}$ ($\sin\theta = 0.01745$) we need $d = 60 \lambda$ approx.
It makes no difference if the wave is longitudinal or transverse. The same formulas apply to both, unless polarisation is involved, because longitudinal waves cannot be polarised.
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