When talking about 'wave-particle duality', you really need to be specific about what is doing the 'waving' and what is being described by the waves.
When you talk about waves in water or waves in a hose, you are talking about mechanical waves; some _thing_ is moving from one place to another. Waves in water are generally 'transverse' waves; the water moves up and down, but the energy carried by the wave moves sideways. The water molecules follow small circular paths, but don't actually move very far; but energy moves along with the wave itself. The energy of the wave is the gravitational potential energy of the water being lifted; as some water drops down it provides energy to lift adjacent water, and the wave travels on.
Sound waves in air are 'longitudinal'; the motion of the air molecules is in the same direction as the energy being transferred. In this case, the air molecules move back and forth, creating regions of high and low pressure. The energy of the wave is carried in the potential energy of this pressure difference, and as this pressure difference collapses in one location the energy released creates a pressure difference nearby; the energy of the wave moves through the air.
The mathematics of how such mechanical waves propagate, and what happens when different wave patterns interact, is well understood, at least in the case of linear media.
Electromagnetic waves are a different phenomena. Rather than having bunches of particles move side to side, what changes is the intensity of the electric field in a region of space. Imagine a simple parallel plate capacitor, connected to an AC source. The plates don't move at all, but the electric field between the plates is constantly changing. The electric field is not moving side to side; it is simply _there_ with a changing value.
However a changing electric field will create a magnetic field, just as a current will create a magnetic field. With the capacitor above (and its changing electric field) we get a changing magnetic field as well. The magnetic field doesn't move from side to side, it is simply _there_ with a changing value.
On top of this, a changing magnetic field will create an electric field. This chain gives us our mechanism for energy transfer.
A bit of electric field in free space, with no charges to maintain it, is analogous to a pile of water in the middle of the ocean without a container to hold it. The pile of water wants to fall down to sea level, but the potential energy needs to go somewhere; in the case of the ocean some of that energy will go into lifting other water, the net result being that the pile of water appears to move; you have a wave. In the case of the electric field, the collapsing electric field creates a magnetic field which creates an electric field, and the 'burp' in the electric field appears to move.
The 'wave' nature of electromagnetic waves, and the 'wave' nature of ocean waves come from very different basic physics. The reason that both are 'waves' is that the same mathematics describes the propagation of these waves, the interference of the waves, etc.
Yet another place that waves show up is in the quantum mechanical description of the locations of particles. An electron is not 'waving' from side to side, but we cannot say _exactly_ where an electron is and how it is moving. The best possible description of the location of an electron is a probability function, and it turns out that these probability functions are described by the same sort of mathematics used to describe the shapes and interactions of waves. So we say that the position of the electron is described by a 'wavefunction', and additionally when electrons interact we find that those interactions are governed by the same sort of mathematics that we see when different mechanical waves interact.
I'm going to stop here because I've overstepped my sure knowledge and because this field is better explained by others who know more. My only basic point is that you shouldn't take the concept of 'wave' too far; ocean waves, sound waves, electromagnetic waves, and electron position waves are all different things that happen to be described _in part_ by similar mathematics.
-Jon
