Can elementary particles be explained adequately by a wave-only model? I have been watching quantum mechanics documentaries and reading a layman's book called "The Quantum Universe".  I believe I understand why the double slit experiments exclude a particle only model.  However I do not understand why the particle portion of particle-wave duality is needed.  When I google the title to this question I do not get an adequate explanation of why the particle side of wave-particle duality is needed I feel.  I believe the explanations assert that a particle moves in a wave-like/probabilistic manner but what is the evidence that requires a particle even exist instead of the wave itself being the whole story?
Is it because elementary particles have quantized states?  Can elementary 'waves' not simply exist in quantized states without a particle?  I guess I would also like to know how a wave-only model would differ from string theory if you would not mind.  My understanding is that string theory replaces particles with vibrating strings that seem an awful lot like quantized waves in my head.  
Forgive me if this is duplicate, my googlefu did not reveal a duplicate.
 A: 
However I do not understand why the particle portion of particle-wave duality is needed.

One of the theories which fails if light was considered to be made of just waves is the photoelectric effect. Had light been just a wave, the energy of the light wave must be proportional to the brightness and hence the wave's amplitude. We could base our predictions as:

*

*The kinetic energy of emitted photoelectrons should increase with the light wave's amplitude.

*With increasing frequency of incident light, the rate of electron emission reflected in the photocurrent should also increase.

However, this is completely at odds with what we observe:

*

*The kinetic energy of photoelectrons increases with light frequency. It remains constant as light amplitude increases.

*Electric current remains constant as light frequency increases. It increases with light amplitude.

*Regardless of the amplitude, a light frequency too low does not produce any photocurrent -- this feature immediately implies a discrete version of energy transfer (thanks to J.G.).

The wave theory completely fails. What we hypothesize instead is that light comes as discrete particles with a defined energy - photons. That fits both the sets of observations.
The photoelectric effect introduced evidence that light exhibited particle properties on the quantum scale of atoms. At least, light can achieve a sufficient localization of energy to eject an electron from a metal surface.
A: 
Can elementary particles be explained adequately by a wave-only model?

The answer is no, not  with the mathematical tools we have developed so far to describe data and observations. 
The  electron is the first elementary particle observed experimentally.
This single electron at a time double slit experiment

shows that single electrons exist ( frame $a$ with a footprint explainable as for a classical "particle"). It is the accumulation of electrons with the same boundary conditions that builds up a pattern of interference that imposes the need for a wave description in describing what an electron  interacts as, that a particle attribute is not enough to explain the data. The probability hypothesis for quantum mechanical interactions, i.e. for the dimensions consistent with the Heisenberg Uncertainty Principe, is the way to explain interactions of elementary particles, and thus the particles themselves.
It is also good to contemplate this experimental picture as this bubble chamber picture

It shows beam particles entering from below , and one interaction with many particles coming out. We call them particles because their trace is the trace of a particle ,not a wave. The experiment has studied a large accumulation of such interactions, which will show the quantum mechanical interaction under study. 
{ I am partly guessing  it is a study  $K^-$ interaction at 10 GeV/c in the 2 meter hydrogen bubble chamber.  In this photo the curling track in the magnetic field is either a $π^+$, (or  a $K^+$ the ionisation will distinguish the two masses but as there are only four charged tracks    the incoming must be negative)  which decays into a $μ^+$ and a neutrino, andfinally a positron with the accompanying neutrinos/ antineutrinos which cannot be seen }
In the mainstream mathematical model we have developed up to now  assuming a wave nature for single particles is not possible. The wave nature appears in probability distributions, accumulation of data in the same boundary conditions,  single particles behave as classical particles macroscopically.
There exist off the main stream theories and efforts to explain with a deterministic model the quantum mechanical probabilistic nature. One is Bohmian mechanics, but it cannot describe all observations , (it is using an underlying wave description). There are people still working on these lines,  but have not been able to explain all the observations and data that the mainstream theory does.
A: Yes and no. 
The standard model, a quantum field theory, is the most complete description of particles and their interactions. Though physicists don't normally think about it quite like this (e.g. see G. Smith's answer), a quantum field theory is simply the theory of high dimensional fields with quantized excitations (i.e. wave packets) that can exist in superposition. A quantized wave acts like a classical particle on length scales that are much larger than the wavelength. So "yes" in the sense that, at the foundation, the theory is a wave only theory. 
However, one could argue that particles are necessary due the fact that the the wave excitations are quantized, and further that the interaction of the quanta can be discrete, e.g. in particle creation and annihilation. Feynman thought of particles as "any bits of energy that come in lumps." Under that definition, the lumpyness of the field excitations and their interactions make the particles abstraction essential. But still, ontologically, we think of the particles themselves as wave packets of the fields in the modern view.
A: Elementary particles are understood today as the quanta of quantum fields. The fields are ontologically primary and exist even when there are no particles, but a quantum field is not “a wave-only model” as is, say, a classical electromagnetic field. 
Instead, a quantum field is a continuous field, existing everywhere in spacetime, of operators that create and destroy quanta with particle-like properties. Quantum fields are not just waves, nor just particles, but rather a mathematical hybrid for which our classical environment gives us no intuition. Fortunately, mathematics makes them understandable to some degree and we find that models using quantum fields, such as the Standard Model, are extremely accurate.
One single quantum field, extending throughout the universe, can explain all electrons and positrons. (Why are all electrons identical? Because they are quanta of the same field!) One more field can explain all photons. One more can explain all up quarks and antiquarks, etc. A mere seventeen quantum fields, interacting with each other, make up the current Standard Model and are the basis for the world we see, except for gravity.
A: No. As anna v has explained we only ever observe particles. We do not observe waves, but we use them to calculate the probabilities for where particles might be found. Actually, they are not even waves. To take the simplest case, the "wave function" for a plane wave state is actually a helix in a complex valued configuration space. The helix rotates in time, creating the illusion of waves on the real and imaginary axes.

The reason for apparent wave effects is buried deep in the mathematics of probability theory and Hilbert space, and involves a higher level of mathematics than is commonly taught to physicists. However, it does make completely clear that there cannot be any physical wave, or indeed any physical field. This is just how calculations have to be performed in order to preserve the probability interpretation for a system in which probabilities result from indeterminacy, not from unknowns, or hidden variables.
I have given a full conceptual discussion in  Light After Dark II: The Large and the Small and a rigorous mathematical treatment in Light After Dark III: The Mathematics of Gravity and Quanta
A: Sort of, yes.
The many-worlds interpretation of quantum mechanics essentially says that there aren't actually any particles, just the quantum waves and our observations of them - the "particles" are just our limited observations of a small slice of the complete quantum waveform. As a result, you could say that they're a wave-only explanation of fundamental particles, since the particles don't "actually" exist.
Here's a Youtube video explaining it in a bit more detail.
