Question Regarding Hydroacoustics

This is not limited to acoustics. It is any wave phenomenon. But it isn't completely true the way you state it. What is true is that the larger the aperture relative to a wavelength, the more directional the transmitted beam.

An acoustic example. A traditional hi-fi loudspeaker has woofer, mid-range and tweeter, and each is smaller than the last. Ideally each broadcasts the same pattern, but because the wavelength gets smaller, the aperture does as well.

Same for antennas. The bigger the dish, for a given wavelength, the higher the gain, or directivity. Think about satellite dishes back in the 1980s vs. today. The difference is the higher frequency (smaller wavelength) used in today's satellite transmissions.

Now here's the why. It will help if you took high school physics and remember ripple tanks, or if you learned Huygen's Wavefront Principle. I'm going to assume neither and state Huygen's Wavefront Principle and the simple drawing you can do to understand the relationship of aperture, wavelength, and directivity.

Huygens said that the wavefront of a traveling wave acts as the source of the next wavefront, with each point along the wavefront the center of a circle (or sphere) with a radius equal to the wavelength. I don't know how to draw in Cr4, so here is a web site with a perfect drawing:

http://www.mathpages.com/home/kmath242/kmath242.htm

So imagine you take a piece of paper and compass, and draw a small circle at the middle of the page using a compass. Now place the point of the compass on any point of the circle you just drew, and draw another circle. then move the compass point a small distance along the circumference of the first circle, and draw another circle. Repeat until you have another circle of radius twice that of the original. This is how a wavefront propagates through space. It is no different than dropping a rock in a pond. If you continue to repeat this algorithm, you generate a bigger and bigger circle. Imagine your circle now occupies about half the height of the page, so that one quarter of the page is left above and below your circle.

Draw two horizontal lines across the page one quarter of the way down from the top, and one quarter of the way up from the bottom. These two lines are tangential to the top and bottom of the circle. Now erase a section of each line at the center where it touches the circle. At the top of the page, erase a section of the line that is long compared to the radius of the successive small circles you have been drawing. At the bottom, erase a smaller section that is short relative to the small circle radius.

Now draw the next wavefront that will penetrate the slots. At the top, with the large slot, what you will see is that at the very corners of the slot, the wavefront diverges around the edge and starts to spread a little, but in the middle, the wavefront is pretty much unaffected by the slot being there at all, and the wave pretty much goes along the way it was before.

At the bottom, the small slot will cause divergence of the entire transmitted wavefront. It looks as if the small slot is the source of a brand new circular wavefront.

And that's pretty much it. I recall reading a book on submarine sonar, and they explained in there that a sonar dish roughly the same size as a radar dish on an airplane had the same directivity as the radar dish. (Directivity meaning the degree to which the wavefront is a beam vs. spreading out.) The frequencies were slightly different, however: the aircraft radar was X-band (8-12 GHz), while the sonar signal was at 50 kHz. But it works out that the wavelengths were equal. The wavelength of any wave is the wave speed divided by frequency. For radar, the speed is the speed of light, 300,000 km/second, while for sound it is 1500 meters per second (in water). So the wavelengths are indeed the same: 3 cm.

So across a frequency ratio of 50,000 times, the directivity of the two dishes is precisely the same. That is a pretty powerful demonstration of the fundamental nature of wave mechanics and our understanding of it.